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Classical Electrodynamics
by JOHN DAVID JACKSON
Professor of Physics, University of Illinois
CLASSICAL
ELECTRODYNAMICS
John Wiley & Sons, Inc., New York • London • Sydney
ii febiaxov
20 19 18 17 16 15 14
Copyright © 1962 by John Wiley & Sons, Inc. Ail rights reserved.
This book or any part thereof must not
be reproduced in any form without the
written permission of the publisher.
Printed in the United States of America
Library of Congress Catalog Card Number: 628774
ISBN 471 43131 1
To the memory of my father,
Walter David Jackson
Preface
Classical electromagnetic theory, together with classical and quan
tum mechanics, forms the core of presentday theoretical training for
undergraduate and graduate physicists. A thorough grounding in these
subjects is a requirement for more advanced or specialized training.
Typically the undergraduate program in electricity and magnetism
involves two or perhaps three semesters beyond elementary physics, with
the emphasis on the fundamental laws, laboratory verification and elabora
tion of their consequences, circuit analysis, simple wave phenomena, and
radiation. The mathematical tools utilized include vector calculus,
ordinary differential equations with constant coefficients, Fourier series,
and perhaps Fourier or Laplace transforms, partial differential equations,
Legendre polynomials, and Bessel functions.
As a general rule a twosemester course in electromagnetic theory is
given to beginning graduate students. It is for such a course that my book
is designed. My aim in teaching a graduate course in electromagnetism is
at least threefold. The first aim is to present the basic subject matter as a
coherent whole, with emphasis on the unity of electric and magnetic
phenomena, both in their physical basis and in the mode of mathematical
description. The second, concurrent aim is to develop and utilize a number
of topics in mathematical physics which are useful in both electromagnetic
theory and wave mechanics. These include Green's theorems and Green's
functions, orthonormal expansions, spherical harmonics, cylindrical and
spherical Bessel functions. A third and perhaps most important pur
pose is the presentation of new material, especially on the interaction of
VIII
Preface
relativistic charged particles with electromagnetic fields. In this last area
personal preferences and prejudices enter strongly. My choice of topics is
governed by what I feel is important and useful for students interested in
theoretical physics, experimental nuclear and highenergy physics, and that
as yet illdefined field of plasma physics.
The book begins in the traditional manner with electrostatics. The first
six chapters are devoted to the development of Maxwell's theory of
electromagnetism. Much of the necessary mathematical apparatus is con
structed along the way, especially in Chapters 2 and 3, where boundary
value problems are discussed thoroughly. The treatment is initially in
terms of the electric field E and the magnetic induction B, with the derived
macroscopic quantities, D and H, introduced by suitable averaging over
ensembles of atoms or molecules. In the discussion of dielectrics, simple
classical models for atomic polarizability are described, but for magnetic
materials no such attempt is made. Partly this omission was a question of
space, but truly classical models of magnetic susceptibility are not possible.
Furthermore, elucidation of the interesting phenomenon of ferromagnetism
needs almost a book in itself.
The next three chapters (79) illustrate various electromagnetic pheno
mena, mostly of a macroscopic sort. Plane waves in different media,
including plasmas, as well as dispersion and the propagation of pulses, are
treated in Chapter 7. The discussion of wave guides and cavities in Chapter
8 is developed for systems of arbitrary cross section, and the problems of
attenuation in guides and the Q of a cavity are handled in a very general
way which emphasizes the physical processes involved. The elementary
theory of multipole radiation from a localized source and diffraction
occupy Chapter 9. Since the simple scalar theory of diffraction is covered
in many optics textbooks, as well as undergraduate books on electricity and
magnetism, I have presented an improved, although still approximate,
theory of diffraction based on vector rather than scalar Green's theorems'
The subject of magnetohydrodynamics and plasmas receives increasingly
more attention from physicists and astrophysicists. Chapter 10 represents
a survey of this complex field with an introduction to the main physical
ideas involved.
The first nine or ten chapters constitute the basic material of classical
electricity and magnetism. A graduate student in physics may be expected
to have been exposed to much of this material, perhaps at a somewhat
lower level, as an undergraduate. But he obtains a more mature view of it,
understands it more deeply, and gains a considerable technical ability in
analytic methods of solution when he studies the subject at the level of this
book. He is then prepared to go on to more advanced topics. The
advanced topics presented here are predominantly those involving the
Preface ix
interaction of charged particles with each other and with electromagnetic
fields, especially when moving relativistically.
The special theory of relativity had its origins in classical electrodynamics.
And even after almost 60 years, classical electrodynamics still impresses
and delights as a beautiful example of the co variance of physical laws under
Lorentz transformations. The special theory of relativity is discussed in
Chapter 11, where all the necessary formal apparatus is developed, various
kinematic consequences are explored, and the covariance of electrodynamics
is established. The next chapter is devoted to relativistic particle kine
matics and dynamics. Although the dynamics of charged particles in
electromagnetic fields can properly be considered electrodynamics, the
reader may wonder whether such things as kinematic transformations of
collision problems can. My reply is that these examples occur naturally
once one has established the four vector character of a particle's momentum
and energy, that they serve as useful practice in manipulating Lorentz
transformations, and that the end results are valuable and often hard to
find elsewhere.
Chapter 13 on collisions between charged particles emphasizes energy
loss and scattering and develops concepts of use in later chapters. Here
for the first time in the book I use semiclassical arguments based on the
uncertainty principle to obtain approximate quantummechanical ex
pressions for energy loss, etc., from the classical results. This approach, so
fruitful in the hands of Niels Bohr and E. J. Williams, allows one to see
clearly how and when quantum mechanical effects enter to modify classical
considerations.
The important subject of emission of radiation by accelerated point
charges is discussed in detail in Chapters 14 and 15. Relativistic effects
are stressed, and expressions for the frequency and angular dependence of
the emitted radiation are developed in sufficient generality for all appli
cations. The examples treated range from synchrotron radiation to
bremsstrahlung and radiative beta processes. Cherenkov radiation and the
Weizsacker Williams method of virtual quanta are also discussed. In the
atomic and nuclear collision processes semiclassical arguments are again
employed to obtain approximate quantummechanical results. I lay con
siderable stress on this point because I feel that it is important for the
student to see that radiative effects such as bremsstrahlung are almost
entirely classical in nature, even though involving smallscale collisions.
A student who meets bremsstrahlung for the first time as an example of a
calculation in quantum field theory will not understand its physical basis.
Multipole fields form the subject matter of Chapter 16. The expansion
of scalar and vector fields in spherical waves is developed from first
principles with no restrictions as to the relative dimensions of source and
x Preface
wavelength. Then the properties of electric and magnetic multipole radia
tion fields are considered. Once the connection to the multipole moments
of the source has been made, examples of atomic and nuclear multipole
radiation are discussed, as well as a macroscopic source whose dimensions
are comparable to a wavelength. The scattering of a plane electromagnetic
wave by a spherical object is treated in some detail in order to illustrate a
boundary value problem with vector spherical .waves.
In the last chapter the difficult problem of radiative reaction is discussed.
The treatment is physical, rather than mathematical, with the emphasis on
delimiting the areas where approximate radiative corrections are adequate
and on finding where and why existing theories fail. The original Abraham
Lorentz theory of the selfforce is presented, as well as more recent classical
considerations.
The book ends with an appendix on units and dimensions and a biblio
graphy. In the appendix I have attempted to show the logical steps
involved in setting up a system of units, without haranguing the reader as
to the obvious virtues of my choice of units. I have provided two tables
which I hope will be useful, one for converting equations and symbols and
the other for converting a given quantity of something from so many
Gaussian units to so many mks units, and vice versa. The bibliography
lists books which I think the reader may find pertinent and useful for
reference or additional study. These books are referred to by author's
name in the reading lists at the end of each chapter.
This book is the outgrowth of a graduate course in classical electro
dynamics which I have taught off and on over the past eleven years, at both
the University of Illinois and McGill University. I wish to thank my
colleagues and students at both institutions for countless helpful remarks
and discussions. Special mention must be made of Professor P. R. Wallace
of McGill, who gave me the opportunity and encouragement to teach what
was then a rather unorthodox course in electromagnetism, and Professors
H. W. Wyld and G. Ascoli of Illinois, who have been particularly free with
many helpful suggestions on the treatment of various topics. My thanks
are also extended to Dr. A. N. Kaufman for reading and commenting on a
preliminary version of the manuscript, and to Mr. G. L. Kane for his
zealous help in preparing the index.
J. D. Jackson
Urbana, Illinois
January, 1962
Contents
chapter 1. Introduction to Electrostatics
1.1 Coulomb's law, 1 .
1.2 Electric field, 2.
1.3 Gauss's law, 4.
1 .4 Differential form of Gauss' s law, 6.
1.5 Scalar potential, 7.
1.6 Surface distributions of charges and dipoles, 9.
1.7 Poisson's and Laplace's equations, 12.
1.8 Green's theorem, 14.
1.9 Uniqueness theorem, 15.
1.10 Formal solution of boundaryvalue problem, Green's functions, 18.
1.11 Electrostatic potential energy, 20.
References and suggested reading, 23.
Problems, 23.
chapter 2. BoundaryValue Problems in Electrostatics, I 26
2.1 Method of images, 26.
2.2 Point charge and a grounded conducting sphere, 27.
2.3 Point charge and a charged, insulated, conducting sphere, 31.
2.4 Point charge and a conducting sphere at fixed potential, 33.
2.5 Conducting sphere in a uniform field, 33.
2.6 Method of inversion, 35.
2.7 Green's function for a sphere, 40.
xii Contents
2.8 Conducting sphere with hemispheres at different potentials, 42.
2.9 Orthogonal functions and expansions, 44.
2.10 Separation of variables in rectangular coordinates, 47.
References and suggested reading, 50.
Problems, 51.
chapter 3. BoundaryValue Problems in Electrostatics, II 54
3.1 Laplace's equation in spherical coordinates, 54.
3.2 Legendre polynomials, 56.
3.3 Boundaryvalue problems with azimuthal symmetry, 60.
3.4 Spherical harmonics, 64.
3.5 Addition theorem for spherical harmonics, 67.
3.6 Cylindrical coordinates, Bessel functions, 69.
3.7 Boundaryvalue problems in cylindrical coordinates, 75.
3.8 Expansion of Green's functions in spherical coordinates, 77.
3.9 Use of spherical Green's function expansion, 81.
3.10 Expansion of Green's functions in cylindrical coordinates, 84.
3.11 Eigenfunction expansions for Green's functions, 87.
3.12 Mixed boundary conditions, charged conducting disc, 89.
References and suggested reading, 93.
Problems, 94.
chapter 4. Multipoles, Electrostatics of Macroscopic Media,
Dielectrics 98
4.1 Multipole expansion, 98.
4.2 Multipole expansion of the energy of a charge distribution in an
external field, 101.
4.3 Macroscopic electrostatics, 103.
4.4 Simple dielectrics and boundary conditions, 108.
4.5 Boundaryvalue problems with dielectrics, 1 10.
4.6 Molecular polarizability and electric susceptibility, 116.
4.7 Models for molecular polarizability, 119.
4.8 Electrostatic energy in dielectric media, 123.
References and suggested reading, 127.
Problems, 128.
chapter 5. Magnetostatics 132
5.1 Introduction and definitions, 132.
5.2 Biot and Savart law, 133.
5.3 Differential equations of magnetostatics, Ampere's law, 137.
5.4 Vector potential, 139.
5.5 Magnetic induction of a circular loop of current, 141.
5.6 Localized current distribution, magnetic moment, 145.
Contents xiii
5.7 Force and torque on localized currents in an external field, 148.
5.8 Macroscopic equations, 1 50.
5.9 Boundary conditions, 154.
5.10 Uniformly magnetized sphere, 156.
5.11 Magnetized sphere in an external field, permanent magnets, 160.
5.12 Magnetic shielding, 162.
References and suggested reading, 164.
Problems, 165.
chapter 6. Time Varying Fields, Maxwell's Equations, Con
servation Laws 169
6.1 Faraday's law of induction, 170.
6.2 Energy in the magnetic field, 173.
6.3 Maxwell's displacement current, Maxwell's equations, 177.
6.4 Vector and scalar potentials, wave equations, 179.
6.5 Gauge transformations, 181.
6.6 Green's function for the timedependent wave equation, 183.
6.7 Initial value problem, Kirchhoff's integral representation, 186.
6.8 Poynting's theorem, 189.
6.9 Conservation laws, 190.
6.10 Macroscopic equations, 194.
References and suggested reading, 198.
Problems, 198.
chapter 7. Plane Electromagnetic Waves 202
7.1 Plane waves in a nonconducting medium, 202.
7.2 Linear and circular polarization, 205.
7.3 Superposition of waves, group velocity, 208.
7.4 Propagation of a pulse in a dispersive medium, 212.
7.5 Reflection and refraction, 216.
7.6 Polarization by reflection, total internal reflection, 220.
7.7 Waves in a conducting medium, 222.
7.8 Simple model for conductivity, 225.
7.9 Transverse waves in a tenuous plasma, 226.
References and suggested reading, 231.
Problems, 231.
chapter 8. Wave Guides and Resonant Cavities 235
8.1 Fields at the surface of and within a conductor, 236.
8.2 Cylindrical cavities and wave guides, 240.
8.3 Wave guides, 244.
8.4 Modes in a rectangular wave guide, 246.
8.5 Energy flow and attenuation in wave guides, 248.
civ Contents
8.6 Resonant cavities, 252.
8.7 Power losses in a cavity, 255.
8.8 Dielectric wave guides, 259.
References and suggested reading, 264.
Problems, 264.
chapter 9. Simple Radiating Systems and Diffraction 268
9.1 Fields and radiation of a localized source, 268.
9.2 Oscillating electric dipole, 271.
9.3 Magnetic dipole and quadrupole fields, 273.
9.4 Centerfed linear antenna, 277.
9.5 Kirchhoff's integral for diffraction, 280.
9.6 Vector equivalents of Kirchhoff's integral, 283.
9.7 Babinet's principle, 288.
9.8 Diffraction by a circular aperture, 292.
9.9 Diffraction by small apertures, 297.
9.10 Scattering by a conducting sphere at short wavelengths, 299.
References and suggested reading, 304.
Problems, 305.
chapter 10. M agnetohydrodynamics and Plasma Physics 309
10.1 Introduction and definitions, 309.
10.2 Magnetohydrodynamic equations, 311.
10.3 Magnetic diffusion, viscosity, and pressure, 313.
10.4 Magnetohydrodynamic flow, 316.
10.5 Pinch effect, 320.
10.6 Dynamic model of the pinch effect, 322.
10.7 Instabilities, 326.
10.8 Magnetohydrodynamic waves, 329.
10.9 Highfrequency plasma oscillations, 335.
10.10 Shortwavelength limit, Debye screening distance, 339.
References and suggested reading, 343.
Problems, 343.
chapter 11. Special Theory of Relativity 347
11.1 Historical background and key experiments, 347.
11.2 Postulates of special relativity, Lorentz transformation, 352.
11.3 FitzGeraldLorentz contraction and time dilatation, 357.
11.4 Addition of velocities, Doppler shift, 360.
11.5 Thomas precession, 364.
11.6 Proper time and light cone, 369.
11.7 Lorentz transformations as orthogonal transformations, 371.
11.8 4vectors and tensors, 374.
Contents xv
11.9 Covariance of electrodynamics, 377.
11.10 Transformation of electromagnetic fields, 380.
11.11 Covariance of the force equation and the conservation laws, 383.
References and suggested reading, 386.
Problems, 387.
chapter 12. RelativisticParticle Kinematics and Dynamics 391
12.1 Momentum and energy of a particle, 391.
12.2 Kinematics of decay of an unstable particle, 394.
12.3 Center of momentum transformation, 397.
12.4 Transformation of momenta from the center of momentum frame
to the laboratory, 400.
12.5 Covariant Lorentz force equation, Lagrangian and Hamiltonian,
404.
12.6 Relativistic corrections to the Lagrangian for interacting charged
particles, 409.
12.7 Motion in a uniform, static, magnetic field, 411.
12.8 Motion in combined uniform, static, electric and magnetic fields,
412.
12.9 Particle drifts in nonuniform magnetic fields, 415.
12.10 Adiabatic invariance of flux through an orbit, 419.
References and suggested reading, 424.
Problems, 425.
chapter 13. Collisions between Charged Particles, Energy Loss,
and Scattering 429
13.1 Energy transfer in a Coulomb collision, 430.
13.2 Energy transfer to a harmonically bound charge, 434.
13.3 Classical and quantummechanical energy loss, 438.
13.4 Density effect in collision energy loss, 443.
13.5 Energy loss in an electronic plasma, 450.
13.6 Elastic scattering of fast particles by atoms, 451.
13.7 Mean square angle of scattering, multiple scattering, 456.
13.8 Electrical conductivity of a plasma, 459.
References and suggested, reading, 462.
Problems, 462.
chapter 14. Radiation by Moving Charges 464
14.1 LienardWiechert potentials and fields, 464.
14.2 Larmor's radiated power formula and its relativistic
generalization, 468.
14.3 Angular distribution of radiation, 472.
14.4 Radiation by an extremely relativistic charged particle, 475.
xvi Contents
14.5 General angular and frequency distributions of radiation from
accelerated charges, 477.
14.6 Frequency spectrum from relativistic charged particle in an instan
taneously circular orbit, synchrotron radiation, 481.
14.7 Thomson scattering, 488.
14.8 Scattering by quasifree charges, 491.
14.9 Cherenkov radiation, 494.
References and suggested reading, 499.
Problems, 500.
chapter 15. Bremsstrahlung, Method of Virtual Quanta, Radia
tive Beta Processes 505
15.1 Radiation emitted during collisions, 506.
15.2 Bremsstrahlung in nonrelativistic Coulomb collisions, 509.
15.3 Relativistic bremsstrahlung, 513.
15.4 Screening, relativistic radiative energy loss, 516.
15.5 WeizsackerWilliams method of virtual quanta, 520.
15.6 Bremsstrahlung as the scattering of virtual quanta, 525.
15.7 Radiation emitted during beta decay, 526.
15.8 Radiation emitted in orbitalelectron capture, 528.
References and suggested reading, 533.
Problems, 534.
chapter 16. Multipole Fields 538
16.1 Scalar spherical waves, 538.
16.2 Multipole expansion of electromagnetic fields, 543.
16.3 Properties of multipole fields, energy and angular momentum of
radiation, 546.
16.4 Angular distributions, 550.
16.5 Sources of multipole radiation, multipole moments, 553.
16.6 Multipole radiation in atoms and nuclei, 557.
16.7 Radiation from a linear, centerfed antenna, 562.
16.8 Spherical expansion of a vector plane wave, 566.
16.9 Scattering by a conducting sphere, 569.
16.10 Boundary value problems with multipole fields, 574.
References and suggested reading, 574.
Problems, 574.
chapter 17. Radiation Damping, SelfFields of a Particle,
Scattering and Absorption of Radiation by a Bound
System 578
17.1 Introductory considerations, 578.
17.2 Radiative reaction force, 581.
Contents xvu
17.3 AbrahamLorentz evaluation of the selfforce, 584.
17.4 Difficulties with the AbrahamLorentz model, 589.
17.5 Lorentz transformation properties of the AbrahamLorentz model,
Poincare stresses, 590.
17.6 Covariant definitions of selfenergy and momentum, 594.
17.7 Integrodifferential equation of motion, including damping, 597.
17.8 Line breadth and level shift of an oscillator, 600.
17.9 Scattering and absorption of radiation by an oscillator, 602.
References and suggested reading, 607.
Problems, 608.
appendix. Units and Dimensions 611
Bibliography 622
Index 625
Introduction to
Electrostatics
Although amber and lodestone were known by the ancient Greeks,
electrodynamics developed as a quantitative subject in about 80 years.
Coulomb's observations on the forces between charged bodies were made
around 1785. About 50 years later, Faraday was studying the effects of
currents and magnetic fields. By 1864, Maxwell had published his famous
paper on a dynamical theory of the electromagnetic field.
We will begin our discussion with the subject of electrostatics— problems
involving timeindependent electric fields. Much of the material will be
covered rather rapidly because it is in the nature of a review. We will use
electrostatics as a testing ground to develop and use mathematical tech
niques of general applicability.
1.1 Coulomb's Law
All of electrostatics stems from the quantitative statement of Coulomb's
law concerning the force acting between charged bodies at rest with respect
to each other. Coulomb (and, even earlier, Cavendish) showed experi
mentally that the force between two small charged bodies separated a
distance large compared to nheir dimensions
(1) varied directly as the magnitude of each charge,
(2) varied inversely as the square of the distance between them,
(3) was directed along the line joining the charges,
(4) was attractive if the bodies were oppositely charged and repulsive
if the bodies had the same type of charge.
Furthermore it was shown experimentally that the total force produced
1
2 Classical Electrodynamics
on one small charged body by a number of the other small charged bodies
placed around it was the vector sum of the individual twobody forces of
Coulomb.
1.2 Electric Field
Although the thing that eventually gets measured is a force, it is useful
to introduce a concept one step removed from the forces, the concept of
an electric field due to some array of charged bodies. At the moment, the
electric field can be defined as the force per unit charge acting at a given
point. It is a vector function of position, denoted by E. One must be
careful in its definition, however. It is not necessarily the force that one
would observe by placing one unit of charge on a pith ball and placing it
in position. The reason is that one unit of charge (e.g., 100 strokes of cat's
fur on an amber rod) may be so large that its presence alters appreciably
the field configuration of the array. Consequently one must use a limiting
process whereby the ratio of the force on the small test body to the charge
on it is measured for smaller and smaller amounts of charge. Experi
mentally, this ratio and the direction of the force will become constant as
the amount of test charge is made smaller and smaller. These limiting
values of magnitude and direction define the magnitude and direction of the
electric field E at the point in question. In symbols we may write
F = qE (1.1)
where F is the force, E the electric field, and q the charge. In this equation
it is assumed that the charge q is located at a point, and the force and the
electric field are evaluated at that point.
Coulomb's law can be written down similarly. If F is the force on a
point charge q x , located at x lf due to another point charge q 2 , located at
x 2 , then Coulomb's law is
F = fc<h<? 2 [ Xl ~ X2 ) 3 (1.2)
Xi ~ x 2  3
Note that q x and q 2 are algebraic quantities which can be positive or
negative. The constant of proportionality k depends on the system of units
used.
The electric field at the point x due to a point charge q x at the point x x
can be obtained directly:
E(x) = k qi ( * ~ Xl) (1.3)
x  x x  3
as indicated in Fig. 1.1. The constant k is determined by the unit of charge
[Sect. 1.2] Introduction to Electrostatics
P
Fig. 1.1
chosen. In electrostatic units (esu), unit charge is chosen as that charge
which exerts a force of one dyne on an equal charge located one centimeter
away. Thus, with cgs units, k = 1 and the unit of charge is called the
"statcoulomb." In the mks system, k = (4 7 re ) 1 , where e (= 8.854 x
10~ 12 farad/meter) is the permittivity of free space. We will use esu.*
The experimentally observed linear superposition of forces due to many
charges means that we may write the electric field at x due to a system of
point charges q it located at x„ i = 1, 2, . . . , n, as the vector sum:
W *S^ (L4)
If the charges are so small and so numerous that they can be described by
a charge density p(x') [if Aq is the charge in a small volume Ax Ay Az at
the point x', then Aq = p(x') Ax Ay Az], the sum is replaced by an
integral:
E(x) = L(x')fc^^' (1.5)
j x — x r
where d 3 x' = dx' dy' dz' is a threedimensional volume element at x'.
At this point it is worth while to introduce the Dirac delta function. In one
dimension, the delta function, written d{x  a), is a mathematically improper
function having the properties:
(1) d(x — a) = for x ^ a, and
6(x  a) dx = 1 if the region of integration includes x = a, and is zero
otherwise.
The delta function can be given rigorous meaning as the limit of a peaked curve
such as a Gaussian which becomes narrower and narrower, but higher and
higher, in such a way that the area under the curve is always constant. L.
Schwartz's theory of distributions is a comprehensive rigorous mathematical
approach to delta functions and their manipulations.!
* The question of units is discussed in detail in the Appendix.
t A useful, rigorous account of the Dirac delta function is given by Lighthill. (Full
references for items cited in the text or footnotes by author only will be found in the
Bibliography.)
(2)
4 Classical Electrodynamics
From the definitions above it is evident that, for an arbitrary function f(x),
(3) f(x) d(x  a) dx = /(a), and
(4) (f(x) d'(x a)dx= /'(a),
where a prime denotes differentiation with respect to the argument.
If the delta function has as argument a function f(x) of the independent
variable x, it can be transformed according to the rule,
(5) */<*» 2 di
dx l/
d{x  x { ) ,
where /(x) is assumed to have only simple zeros, located at x = a^.
In more than one dimension, we merely take products of delta functions in
each dimension. In three dimensions, for example,
(6) (5(x  X) = d(x x  JT X ) d(x 2  X 2 ) d(x 3  X 3 )
is a function which vanishes everywhere except at x = X, and is such that
,„ v f «., „ (I if A V contains x = X,
(7) <5(x  X) tPx =
Jav [0 if A V does not contain x = X.
Note that a delta function has the dimensions of an inverse volume in whatever
number of dimensions the space has.
A discrete set of point charges can be described with a charge density by
means of delta functions. For example,
n
P (x) = £ fr c5(x  x .) (1>6)
i = l
represents a distribution of n point charges q u located at the points x^. Substitu
tion of this charge density (1.6) into (1.5) and integration, using the properties of
the delta function, yields the discrete sum (1.4).
1.3 Gauss's Law
The integral (1.5) is not the most suitable form for the evaluation of
electric fields. There is another integral result, called Gauss's law, which
is often more useful and which furthermore leads to a differential equation
for E(x). To obtain Gauss's law we first consider a point charge q and a
closed surface S, as shown in Fig. 1.2. Let r be the distance from the
charge to a point on the surface, n be the outwardly directed unit normal
to the surface at that point, da be an element of surface area. If the electric
field E at the point on the surface due to the charge q makes an angle
with the unit normal, then the normal component of E times the area
element is : „^„ fl
_, , cos a , , . _.
E • n da = q — — da (1.7)
r 2
Since E is directed along the line from the surface element to the charge q,
[Sect. 1.3]
Introduction to Electrostatics
q inside S
E n
q outside S
Fig. 1.2 Gauss's law. The normal component of electric field is integrated over the
closed surface S. If the charge is inside (outside) S, the total solid angle subtended at
the charge by the inner side of the surface is 4n (zero).
cos 6 da = r 2 dQ, where dQ. is the element of solid angle subtended by da
at the position of the charge. Therefore
Enda = qdQ. 08)
If we now integrate the normal component of E over the whole surface, it
is easy to see that
„ , (477A if q lies inside S n Q> .
is ^ if q lies outside S
6 Classical Electrodynamics
This result is Gauss's law for a single point charge. For a discrete set of
charges, it is immediately apparent that
E.nda = 4«2ft (1.10)
where the sum is over only those charges inside the surface S. For a
continuous charge density p(x), Gauss's law becomes:
i> E • n da = 4tt\ p (x) d z x (l.H)
where V is the volume enclosed by S.
Equation (1 . 1 1) is one of the basic equations of electrostatics. Note that
it depends upon
(1) the inverse square law for the force between charges,
(2) the central nature of the force,
(3) the linear superposition of the effects of different charges.
Clearly, then, Gauss's law holds for Newtonian gravitational force fields,
with matter density replacing charge density.
It is interesting to observe that before Coulomb's observations
Cavendish, by what amounted to a direct application of Gauss's law, did
an experiment with two concentric conducting spheres and deduced that
the power law of the force was inverse nth power, where n = 2.00 ± 0.02.
By a refinement of the technique, Maxwell showed that n = 2.0 ± 0.00005.
(See Jeans, p. 37, or Maxwell, Vol. 1, p. 80.)
1.4 Differential Form of Gauss's Law
Gauss's law can be thought of as being an integral formulation of the
law of electrostatics. We can obtain a differential form (i.e., a differential
equation) by using the divergence theorem. The divergence theorem states
that for any vector field A(x) defined within a volume V surrounded by
the closed surface 5" the relation
<t A • n da = V • A d 3 x
holds between the volume integral of the divergence of A and the surface
integral of the outwardly directed normal component of A. The equation
in fact can be used as the definition of the divergence (see Stratton, p. 4).
To apply the divergence theorem we consider the integral relation
expressed in Gauss's theorem:
<P E • n da = 4tt p(x) d 3 z
•>s Jv
I
[Sect. 1 .5] Introduction to Electrostatics 7
Now the divergence theorem allows us to write this as :
(V . E  4t77>) d z x = (1.12)
>v
for an arbitrary volume V. We can, in the usual way, put the integrand
equal to zero to obtain
VE = 4*7> (1.13)
which is the differential form of Gauss's law of electrostatics. This
equation can itself be used to solve problems in electrostatics. However,
it is often simpler to deal with scalar rather than vector functions of position,
and then to derive the vector quantities at the end if necessary (see below).
1.5 Another Equation of Electrostatics and the Scalar Potential
The single equation (1.13) is not enough to specify completely the three
components of the electric field E(x). Perhaps some readers know that a
vector field can be specified completely if its divergence and curl are given
everywhere in space. Thus we look for an equation specifying curl E as a
function of position. Such an equation, namely,
V x E = (1.14)
follows directly from our generalized Coulomb's law (1.5):
E(x) == f P (x') ^ ^1 d 3
J H 'lxx'l 3
The vector factor in the integrand, viewed as a function of x, is the negative
gradient of the scalar l/x — x' :
x  x'l 3 \x  x'i/
Since the gradient operation involves x, but not the integration variable x',
it can be taken outside the integral sign. Then the field can be written
E(x) == V ( p(x ' } d z x' (1.15)
J x — x'
Since the curl of the gradient of any scalar function of position vanishes
(V x Vy) = 0, for all xp), (1.14) follows immediately from (1.15).
Note that V x E = depends on the central nature of the force
between charges, and on the fact that the force is a function of relative
distances only, but does not depend on the inverse square nature.
Classical Electrodynamics
Fig. 1.3
In (1.15) the electric field (a vector) is derived from a scalar by the
gradient operation. Since one function of position is easier to deal with
than three, it is worth while concentrating on the scalar function and giving
it a name. Consequently we define the "scalar potential" <D(x) by the
ec l uation: E=VO (1.16)
Then (1.15) shows that the scalar potential is given in terms of the charge
density by C ( '\
0(x)= p{X) d 3 x' (1.17)
J x  x'
where the integration is over all charges in the universe, and <2> is arbitrary
to the extent that a constant can be added to the right side of (1.17).
The scalar potential has a physical interpretation when we consider the
work done on a test charge q in transporting it from one point 04) to
another point (B) in the presence of an electric field E(x), as shown in Fig.
1.3. The force acting on the charge at any point is
F = qE
so that the work done in moving the charge from A to B is
W =  ¥dl= q) Edl (1.18)
The minus sign appears because we are calculating the work done on the
charge against the action of the field. With definition (1.16) the work can
be written » /.
W = q\ VO • d\ = q\ d®= g[0 7j >  OJ (1.19)
which shows that q<& can be interpreted as the potential energy of the test
charge in the electrostatic field.
From (1.18) and (1.19) it can be seen that the line integral of the electric
field between two points is independent of the path and is the negative of
the potential difference between the points:
Vdl= (O^OJ (1.20)
i:
[Sect. 1.6]
Introduction to Electrostatics
This follows directly, of course, from definition (1.16). If the path is closed,
the line integral is zero,
E • d\ =
(1.21)
a result that can also be obtained directly from Coulomb's law. Then
application of Stokes's theorem [if A(x) is a vector field, S is an open
surface, and C is the closed curve bounding S,
(fc A • d\ =
(V x A) • n da
where d\ is a line element of C, n is the normal to S, and the path C is
traversed in a righthand screw sense relative to n] leads immediately back
to V x E = 0.
1.6 Surface Distributions of Charges and Dipoles and Discontinuities
in the Electric Field and Potential
One of the common problems in electrostatics is the determination of
electric field or potential due to a given surface distribution of charges.
Gauss's law (1.11) allows us to write down a partial result directly. If a
surface S, with a unit normal n, has a surfacecharge density of <r(x)
(measured in statcoulombs per square centimeter) and electric fields E x
and E 2 on either side of the surface, as shown in Fig. 1 .4, then Gauss's law
tells us immediately that
(E 2  E x ) • n = 4tto (1.22)
This does not determine E x and E 2 unless there are no other sources of
field and the geometry and form a are especially simple. All that (1.22)
says is that there is a discontinuity of 4tt<j in the normal component of
electric field in crossing a surface with a surfacecharge density a, the
crossing being made from the "inner" to the "outer" side of the surface.
Fig. 1.4 Discontinuity in the normal com
ponent of electric field across a surface layer
of charge.
*E 2
10 Classical Electrodynamics
The tangential component of electric field can be shown to be continuous
across a boundary surface by using (1.21) for the line integral of E around
a closed path. It is only necessary to take a rectangular path with negligible
ends and one side on either side of the boundary.
A general result for the potential (and hence the field, by differentiation)
at any point in space (not just at the surface) can be obtained from (1.17)
by replacing p d z x by a da:
Js
<D(x)= r^hda' (1.23)
>S X — X
Another problem of interest is the potential due to a dipolelayer
distribution on a surface S. A dipole layer can be imagined as being formed
by letting the surface S have a surfacecharge density <j(x) on it, and
another surface S', lying close to S, have an equal and opposite surface
charge density on it at neighboring points, as shown in Fig. 1.5. The
dipolelayer distribution of strength D(x) is formed by letting S' approach
infinitesimally close to S while the surfacecharge density o(x) becomes
infinite in such a manner that the product of a(x) and the local separation
d(x) of S and S' approaches the limit Z>(x) :
lim a(x)d(x) = D(x) (1.24)
The direction of the dipole moment of the layer is normal to the surface S
and in the direction going from negative to positive charge.
To find the potential due to a dipole layer we can consider a single dipole
and then superpose a surface density of them, or we can obtain the same
result by performing mathematically the limiting process described in words
above on the surfacedensity expression (1.23). The first way is perhaps
simpler, but the second gives useful practice in vector calculus. Con
sequently we proceed with the limiting process. With n, the unit normal to
Fig. 1.5 Limiting process involved in
creating a dipole layer.
[Sect. 1.6]
Introduction to Electrostatics
11
Fig. 1.6 Dipolelayer geometry.
the surface S, directed away from S', as shown in Fig. 1.6, the potential
due to the two close surfaces is
«,) = f _•&!. da > _ f — e&o_ da .
Js x — x' Js' x — x' + nd\
For small d we can expand x — x' + nJ _1 . Consider the general
expression x + a _1 , where a < x. Then we write
1
1
x + a sjx 2 + a 2 + 2a • x
x +
a; \a;/
This is, of course, just a Taylor's series expansion in three dimensions. Tn
this way we find that the potential becomes [upon taking the limit (1.24)]:
O(x) = D(x')n • V ( — ) da' (1.25)
■ Js \x — x'/
Equation (1.25) has a simple geometrical interpretation. We note that
rr/j 1 \ j f cos da' ,„
n • V ( da = = — dQ.
\x  x'/ x  x' 2
where dQ, is the element of solid angle subtended at the observation point
by the area element da, as indicated in Fig. 1 .7. Note that dQ, has a positive
sign if is an acute angle, i.e., when the observation point views the "inner"
side of the dipole layer. The potential can be written:
(D(x) =  D(x') dQ
J si
(1.26)
12 Classical Electrodynamics
Fig. 1.7 The potential at P due to the
dipole layer D on the area element da' is
just the negative product of D and the
solid angle element dQ, subtended by da'
at P.
For a constant surfacedipolemoment density D, the potential is just the
product of the moment and the solid angle subtended at the observation
point by the surface, regardless of its shape.
There is a discontinuity in potential in crossing a double layer. This
can be seen by letting the observation point come infinitesimally close to
the double layer. The double layer is now imagined to consist of two
parts, one being a small disc directly under the observation point. The
disc is sufficiently small that it is sensibly flat and has constant surface
dipolemoment density D. Evidently the total potential can be obtained
by linear superposition of the potential of the disc and that of the remain
der. From (1.26) it is clear that the potential of the disc alone has a
discontinuity of AttD in crossing from the inner to the outer side, being
—2ttD on the inner side and \2ttD on the outer. The potential of the
remainder alone, with its hole where the disc fits in, is continuous across
the plane of the hole. Consequently the total potential jump in crossing
the surface is: 0,^ = 4^ (1.27)
This result is analogous to (1.22) for the discontinuity of electric field in
crossing a surfacecharge density. Equation (1.27) can be interpreted
"physically" as a potential drop occurring "inside" the dipole layer, and
can be calculated as the product of the field between the two layers of
surface charge times the separation before the limit is taken.
1.7 Poisson's and Laplace's Equations
In Sections 1.4 and 1.5 it was shown that the behavior of an electro
static field can be described by the two differential equations :
VE = 4tt P (1.13)
and V x E = (1.14)
the latter equation being equivalent to the statement that E is the gradient
of a scalar function, the scalar potential <I> :
E= VO (1.16)
[Sect. 1 .7] Introduction to Electrostatics 13
Equations (1.13) and (1.16) can be combined into one partial differential
equation for the single function O(x) :
V 2 = 4t7 P (1.28)
This equation is called Poisson's equation. In regions of space where there
is no charge density, the scalar potential satisfies Laplace's equation:
V 2 <D = (1.29)
We already have a solution for the scalar potential in expression (1.17):
0(x) = M*2dV (1.17)
J x — X 
To verify that this does indeed satisfy Poisson's equation (1.28) we operate
with the Laplacian on both sides :
V 2 0> = V 2 f£&2_ d z x' = f P (x')V 2 ( — ) d*x' (1.
J x — x' J \x — x'l/
30)
We must now calculate the value of V 2 (l/x — x'). It is convenient (and
allowable) to translate the origin to x' and so consider V 2 (l/r), where r is
the magnitude of x. By direct calculation we find that V 2 (l/r) = for
\r/ r dr 2 \ rl r dr 2
At r = 0, however, the expression is undefined. Hence we must use a
limiting process. Since we anticipate something like a Dirac delta function,
we integrate V 2 (l/r) over a small volume V containing the origin. Then we
use the divergence theorem to obtain a surface integral:
Js dr \ r
r 2 dQ. = 4tt
It has now been established that V 2 (l/r) = for r ^ 0, and that its volume
integral is — 4n. Consequently we can write the improper (but mathe
matically justifiable) equation, V 2 (l/r) = — 4tt«3(x), or, more generally,
\x  x'l/
4t7<5(x  x') (1.31)
Having established the singular nature of the Laplacian of 1/r, we can
now complete our check on (1.17) as a solution of Poisson's equation.
14 Classical Electrodynamics
Equation (1.30) becomes
V 2 0> = f p(x')[47r<5(x  x')] d*x' = 4tt P (x)
verifying the correctness of our solution (1.17).
1.8 Green's Theorem
If electrostatic problems always involved localized discrete or continuous
distributions of charge with no boundary surfaces, the general solution
(1.17) would be the most convenient and straightforward solution to any
problem. There would be no need of Poisson's or Laplace's equation. In
actual fact, of course, many, if not most, of the problems of electrostatics
involve finite regions of space, with or without charge inside, and with
prescribed boundary conditions on the bounding surfaces. These boundary
conditions may be simulated by an appropriate distribution of charges
outside the region of interest (perhaps at infinity), but (1.17) becomes
inconvenient as a means of calculating the potential, except in simple cases
(e.g., method of images).
To handle the boundary conditions it is necessary to develop some new
mathematical tools, namely, the identities or theorems due to George
Green (1824). These follow as simple applications of the divergence
theorem. The divergence theorem :
{
V • A d 3 x = Q> A • n da
v Js
applies to any vector field A defined in the volume V bounded by the closed
surface S. Let A = <f>Vy>, where <f> and \p are arbitrary scalar fields. Now
V . (<jHip) = <f>V 2 y + V<£ • Vy> (l .32)
and
«iV^n=^^ (1.33)
on
where d/dn is the normal derivative at the surface S (directed outwards
from inside the volume V). When (1.32) and (1.33) are substituted into
the divergence theorem, there results Green's first identity:
f UV> + V<£ • Vy>) d z x = <t <f> ^ da (1 .34)
Jv Js on
If we write down (1.34) again with </> and xp interchanged, and then subtract
it from (1.34), the V<£ • Vy> terms cancel, and we obtain Green's second
[Sect. 1.9]
Introduction to Electrostatics
identity or Green's theorem :
I
. dtp deb
dn dnJ
da
15
(1.35)
Poisson's differential equation for the potential can be converted into an
integral equation if we choose a particular yt, namely \jR = l/x — x',
where x is the observation point and x' is the integration variable. Further,
we put $ = <J>, the scalar potential, and make use of V 2 = — Anp. From
(1.31) we know that V 2 (l/rt) == 4tt6(x  x), so that (1.35) becomes
— 4tt(^(x')6(x
X') + ^ />(X')
R
d 3 x' =
•js
L dn'\R/
10®
Rdri.
da'
If the point x lies within the volume V, we obtain:
« x) _ f £&L> ** + L 6, \L $ _ 0. ±(l)] da' (..36)
v ' J v R ATrJslRdn' dn'\R/J
If x lies outside the surface S, the lefthand side of (1.36) is zero. [Note
that this is consistent with the interpretation of the surface integral as being
the potential due to a surfacecharge density a = (l/47r)(d<I>/dfl') and a
dipole layer D = — (1/4tt)0. The discontinuities in electric field and
potential (1.22) and (1.27) across the surface then lead to zero field and
zero potential outside the volume V.]
Two remarks are in order about result (1 .36). First, if the surface S goes
to infinity and the electric field on S falls off faster than R~\ then the
surface integral vanishes and (1.36) reduces to the familiar result (1.17).
Second, for a chargefree volume the potential anywhere inside the volume
(a solution of Laplace's equation) is expressed in (1.36) in terms of the
potential and its normal derivative only on the surface of the volume. This
rather surprising result is not a solution to a boundaryvalue problem, but
only an integral equation, since the specification of both $ and d<D/d«
{Cauehy boundary conditions) is an overspecification of the problem. This
will be discussed in detail in the next sections, where techniques yielding
solutions for appropriate boundary conditions will be developed using
Green's theorem (1.35).
1.9 Uniqueness of the Solution with Dirichlet or Neumann Boundary
Conditions
The question arises as to what are the boundary conditions appropriate
for Poisson's (or Laplace's) equation in order that a unique and well
behaved (i.e., physically reasonable) solution exist inside the bounded
16 Classical Electrodynamics
region. Physical experience leads us to believe that specification of the
potential on a closed surface (e.g., a system of conductors held at different
potentials) defines a unique potential problem. This is called a Dirichlet
problem, or Dirichlet boundary conditions. Similarly it is plausible that
specification of the electric field (normal derivative of the potential) every
where on the surface (corresponding to a given surfacecharge density)
also defines a unique problem. Specification of the normal derivative
is known as the Neumann boundary condition. We now proceed to prove
these expectations by means of Green's first identity (1.34).
We want to show the uniqueness of the solution of Poisson's equation,
V 2 <I> = — 4irp, inside a volume V subject to either Dirichlet or Neumann
boundary conditions on the closed bounding surface S. We suppose, to
the contrary, that there exist two solutions Q) 1 and <J> 2 satisfying the same
boundary conditions. Let
U=® 2 <$> 1 (1.37)
Then V 2 £/ = inside V, and U = or dU/dn = on S for Dirichlet and
Neumann boundary conditions, respectively. From Green's first identity
(1.34), with (f> = \p = U, we find
(UV 2 U +VUVU)d 3 x= <£ U — da (1.38)
Jv Js dn
With the specified properties of U, this reduces (for both types of boundary
conditions) to :
Vl/ 2 d 3 x =
Jv
which implies VU=0. Consequently, inside V, U is constant. For
Dirichlet boundary conditions, U = on 5* so that, inside V, <J\ = d> 2 and
the solution is unique. Similarly, for Neumann boundary conditions, the
solution is unique, apart from an unimportant arbitrary additive constant.
From the righthand side of (1 .38) it is clear that there is also a unique
solution to a problem with mixed boundary conditions (i.e., Dirichlet over
part of the surface S, and Neumann over the remaining part).
It should be clear that a solution to Poisson's equation with both O and
d<&/dn specified on a closed boundary (Cauchy boundary conditions) does
not exist, since there are unique solutions for Dirichlet and Neumann
conditions separately and these will in general not be consistent. The
question of whether Cauchy boundary conditions on an open surface define
a unique electrostatic problem requires more discussion than is warranted
here. The reader may refer to Morse and Feshbach, Section 6.2, pp. 692
706, or to Sommerfeld, Partial Differential Equations in Physics, Chapter
[Sect. 1.9]
Introduction to Electrostatics
17
II, for a detailed discussion of these questions. Morse and Feshbach base
their treatment on the replacement of the partial differential equation by
appropriate difference equations which they then solve by an iterative
procedure. On the other hand, Sommerfeld bases his discussion on the
method of characteristics where possible. The result of these investigations
on which boundary conditions are appropriate is summarized in the table
below (based on one given in Morse and Feshbach), where different types
Type of
Boundary
Condition
Type of Equation
Elliptic:
(Poisson's eq.)
Hyperbolic
(wave eq.)
Parabolic
(heatcon
duction eq.)
—
Dirichlet
Not enough
Too much
Not enough
Too much
Open surface
Not enough
Unique, stable
solution in one
direction
Closed surface
Unique, stable
solution
Too much
Neumann
Not enough
Open surface
Unique, stable
solution in one
direction
Closed surface
Unique, stable
solution in
general
Too much
Cauchy
Open surface
Unphysical
results
Unique, stable
solution
Too much
Closed surface
Too much
Too much
Too much
A stable solution is one for which small changes in the boundary conditions
cause appreciable changes in the solution only in the neighborhood of the
boundary.
of partial differential equations and different kinds of boundary conditions
are listed.
Study of the table shows that electrostatic problems are specified only
by Dirichlet or Neumann boundary conditions on a closed surface (part
or all of which may be at iniinity, of course).
18
Classical Electrodynamics
1.10 Formal Solution of Electrostatic BoundaryValue Problem with
Green's Function
The solution of Poisson's or Laplace's equation in a finite volume V with
either Dirichlet or Neumann boundary conditions on the bounding surface
S can be obtained by means of Green's theorem (1.35) and socalled
"Green's functions."
In obtaining result (1.36)— not a solution— we chose the function y to
be l/x — x', it being the potential of a unit point charge, satisfying the
equation: ,
v ' 2 tbi) = "^ (x " x,) (L31)
The function l/x — x' is only one of a class of functions depending on the
variables x and x', and called Green's functions, which satisfy (1.31). In
V' 2 G(x, x') = 4ttS(x  x') (1.39)
1
general,
where
G(x, x') =
X  X'
+ F(x, x')
(1.40)
with the function F satisfying Laplace's equation inside the volume V:
V 2 F(x, x') =
(1.41)
In facing the problem of satisfying the prescribed boundary conditions
on <D or d$>/dn, we can find the key by considering result (1.36). As has
been pointed out already, this is not a solution satisfying the correct type
of boundary conditions because both <J> and dQ>/dn appear in the surface
integral. It is at best an integral equation for <J>. With the generalized
concept of a Green's function and its additional freedom [via the function
F(x, x')], there arises the possibility that we can use Green's theorem with
tp = G(x, x') and choose F(x, x') to eliminate one or the other of the two
surface integrals, obtaining a result which involves only Dirichlet or
Neumann boundary conditions. Of course, if the necessary <7(x, x')
depended in detail on the exact form of the boundary conditions, the
method would have little generality. As will be seen immediately, this is
not required, and G(x, x') satisfies rather simple boundary conditions on S.
With Green's theorem (1.35), <j> = O, rp = G(x, x'), and the specified
properties of G (1.39), it is simple to obtain the generalization of (1.36):
<D(x) = P (x')G(x, x') dV + — <k
Jv 4n Js
G(x, x ) —   d>(x') v '
on on
da'
(1.42)
[Sect. 1.10] Introduction to Electrostatics 19
The freedom available in the definition of G (1.40) means that we can make
the surface integral depend only on the chosen type of boundary con
ditions. Thus, for Dirichlet boundary conditions we demand :
G D (x,x') = for x' on S (1.43)
Then the first term in the surface integral in (1.42) vanishes and the
solution is
O(x) = f p(x')G D (x, x') d 3 x'  i <£ O(x') ^2 da' (1.44)
Jv 4tt Js on
For Neumann boundary conditions we must be more careful. The
obvious choice of boundary condition on G(x, x') seems to be
^ (x, x') = for x' on S
dn'
since that makes the second term in the surface integral in (1.42) vanish,
as desired. But an application of Gauss's theorem to (1.39) shows that
— da' = —477
sdn'
Consequently the simplest allowable boundary condition on G N is
^(x,x')=— forx'onS (1.45)
dn' K S
where S is the total area of the boundary surface. Then the solution is
O(x) = <<D>s + f p(x)G N (x, x') d*x' + ±£> <^G N da' (1.46)
Jv 477 Js on
where (O)^ is the average value of the potential over the whole surface.
The customary Neumann problem is the socalled "exterior problem" in
which the volume V is bounded by two surfaces, one closed and finite, the
other at infinity. Then the surface area S is infinite; the boundary
condition (1.45) becomes homogeneous; the average value {Q>) s vanishes.
We note that the Green's functions satisfy simple boundary conditions
(1.43) or (1.45) which do not depend on the detailed form of the Dirichlet
(or Neumann) boundary values. Even so, it is often rather involved (if
not impossible) to determine G(x, x') because of its dependence on the
shape of the surface S. We will encounter such problems in Chapter 2
and 3.
The mathematical symmetry property G(x, x') = G(x', x) can be proved
for the Green's functions satisfying the Dirichlet boundary condition
(1.43) by means of Green's theorem with <j> = G(x, y) and xp = G(x\ y),
2® Classical Electrodynamics
where y is the integration variable. Since the Green's function, as a function
of one of its variables, is a potential due to a unit point charge, this sym
metry merely represents the physical interchangeability of the source and
the observation points. For Neumann boundary conditions the symmetry
is not automatic, but can be imposed as a separate requirement.
As a final, important remark we note the physical meaning of F(x, x')>
It is a solution of Laplace's equation inside V and so represents the
potential of a system of charges external to the volume V. It can be
thought of as the potential due to an external distribution of charges so
chosen as to satisfy the homogeneous boundary conditions of zero
potential (or zero normal derivative) on the surface S when combined with
the potential of a point charge at the source point x'. Since the potential
at a point x on the surface due to the point charge depends on the position
of the source point, the external distribution of charge F(x, x') must also
depend on the "parameter" x'. From this point of view, we see that the
method of images (to be discussed in Chapter 2) is a physical equivalent
of the determination of the appropriate F(x, x') to satisfy the boundary
conditions (1.43) or (1.45). For the Dirichlet problem with conductors,
^(x, x') can also be interpreted as the potential due to the surfacecharge
distribution induced on the conductors by the presence of a point charge
at the source point x'.
1.11 Electrostatic Potential Energy and Energy Density
In Section 1.5 it was shown that the product of the scalar potential and
the charge of a point object could be interpreted as potential energy. More
precisely, if a point charge q i is brought from infinity to a point x, in a
region of localized electric fields described by the scalar potential $ (which
vanishes at infinity), the work done on the charge (and hence its potential
energy) is given by
W i =q^>{x i ) (1.47)
The potential can be viewed as produced by an array of (n — 1) charges
VjU — 1,2, • ■ . ,n — 1) at positions x,. Then
nl
; = 1 l^j Xj
so that the potential energy of the charge q t is
(1.48)
W<
*§i7^i c 49 )
[Sect. 1.11] Introduction to Electrostatics 21
It is clear that the total potential energy of all the charges due to all the
forces acting between them is:
n
"2 2^ (1  50)
i = l j<i ' X * X<
as can be seen most easily by adding each charge in succession. A more
symmetric form can be written by summing over / andy unrestricted, and
then dividing by 2 :
w= iyy_is L _ (1 .5i)
It is understood that i = j terms (infinite "selfenergy" terms) are omitted
in the double sum.
For a continuous charge distribution [or, in general, using the Dirac
delta functions (1.6)] the potential energy takes the form:
1 [[pV)PW d * xd * x > (1.52)
2 J J xx'
Another expression, equivalent to (1.52), can be obtained by noting that
one of the integrals in (1.52) is just the scalar potential (1.17). Therefore
W= L(x)0(x)d 3 x (1.53)
Equations (1.51), (1.52), and (1.53) express the electrostatic potential
energy in terms of the positions of the charges and so emphasize the
interactions between charges via Coulomb forces. An alternative, and
very fruitful, approach is to emphasize the electric field and to interpret
the energy as being stored in the electric field surrounding the charges. To
obtain this latter form, we make use of Poisson's equation to eliminate the
charge density from (1.53):
W.= — f(DV 2 <D d z x
877 J
Integration by parts leads to the result:
W = J_ f lV <D 2 d*x =  \ E 2 d 3 x (1.54)
where the integration is over all space. In (1.54) all explicit reference to
charges has gone, and the energy is expressed as an integral of the square
of the electric field over all space. This leads naturally to the identification
of the integrand as an energy density w :
w =  IE! 2 (155)
22 Classical Electrodynamics
j'P
O
Fig. 1.8
This expression for energy density is intuitively reasonable, since regions
of high fields "must" contain considerable energy.
There is perhaps one puzzling thing about (1.55). The energy density is
positive definite. Consequently its volume integral is necessarily non
negative. This seems to contradict our impression from (1.51) that the
potential energy of two charges of opposite sign is negative. The reason
for this apparent contradiction is that (1.54) and (1.55) contain "self
energy" contributions to the energy density, whereas the double sum in
(1.51) does not. To illustrate this, consider two point charges q x and q 2
located at x x and x 2 , as in Fig. 1.8. The electric field at the point P with
coordinate x is
E _ gi(x  x t ) g 2 (x  x 2 )
x  Xl  3 x  x 2  3
so that the energy density (1.55) is
w = 2l! + <?2 2 g iq2 (x  x 2 )  (x  x 2 )
877x Xl  4 8t7xx 2  4 4t7xx 1  3 xx 2  3 K }
Clearly the first two terms are selfenergy contributions. To show that the
third term gives the proper result for the interaction potential energv_we
integrate over all space :
A change of integration variable to p = (x — x^/lxi — x 2  yields
HWp^XLfj^ti^ (1.58)
x x  x 2  Att J P 3 \p + n 3
where n is a unit vector in the direction (x x — x 2 ). By straightforward
integration the dimensionless volume integral can be shown to have the
value 4?r, so that the interaction energy reduces to the expected value.
Forces acting between charged bodies can be obtained by calculating
the change in the total electrostatic energy of the system under small
virtual displacements. Examples of this are discussed in the problems.
Care must be taken to exhibit the energy in a form showing clearly those
[Probs. 1] Introduction to Electrostatics 23
factors which vary with a change in configuration and those which are
kept constant.
As a simple illustration we calculate the force per unit area on the surface
of a conductor with a surfacecharge density <r(x). In the immediate
neighborhood of the surface the energy density is
w = — E 2 = 2t7(7 2 (1.59)
If we now imagine a small outward displacement Ax of an elemental area
Aa of the conducting surface, the electrostatic energy decreases by an
amount which is the product of energy density w and the excluded volume
AxAa: aw= 2tto 2 Aci Ax (1.60)
This means that there is an outward force per unit area equal to Ino 2 = w
at the surface of the conductor. This result is normally derived by taking
the product of the surfacecharge density and the electric field, with care
taken to eliminate the electric field due to the element of surfacecharge
density itself.
REFERENCES AND SUGGESTED READING
On the mathematical side, the subject of delta functions is treated simply but rigor
ously by
Lighthill.
For a discussion of different types of partial differential equations and the appropriate
boundary conditions for each type, see
Morse and Feshbach, Chapter 6,
Sommerfeld, Partial Differential Equations in Physics, Chapter II,
Courant and Hilbert, Vol. II, Chapters IIIVI.
The general theory of Green's functions is treated in detail by
Friedman, Chapter 3,
Morse and Feshbach, Chapter 7.
The general theory of electrostatics is discussed extensively in many of the older books.
Notable, in spite of some oldfashioned notation, are
Maxwell, Vol. 1, Chapters II and IV,
Jeans, Chapters II, VI, VII.
Of more recent books, mention may be made of the treatment of the general theory by
Stratton, Chapter III, and parts of Chapter II.
PROBLEMS
1.1 Use Gauss's theorem to prove the following statements :
(a) Any excess charge placed on a conductor must lie entirely on its
surface. (A conductor by definition contains charges capable of moving
freely under the action of applied electric fields.)
24 Classical Electrodynamics
(b) A closed, hollow conductor shields its interior from fields due to
charges outside, but does not shield its exterior from the fields due to
charges placed inside it.
(c) The electric field at the surface of a conductor is normal to the surface
and has a magnitude 4tto, where a is the charge density per unit area on the
surface.
1.2 Two infinite, conducting, plane sheets of uniform thicknesses t x and t 2 ,
respectively, are placed parallel to one another with their adjacent faces
separated by a distance L. The first sheet has a total charge per unit area
(sum of the surfacecharge densities on either side) equal to q lt while the
second has q 2 . Use symmetry arguments and Gauss's law to prove that
(a) the surfacecharge densities on the adjacent faces are equal and
opposite;
(b) the surfacecharge densities on the outer faces of the two sheets are
the same;
(c) the magnitudes of the charge densities and the fields produced are
independent of the thicknesses t x and t 2 and the separation L.
Find the surfacecharge densities and fields explicitly in terms of q x and
q 2 , and apply your results to the special case q x = —q 2 = Q.
1.3 Each of three charged spheres of radius a, one conducting, one having a
uniform charge density within its volume, and one having a spherically
symmetric charge density which varies radially as r n (n > —3), has a total
charge Q. Use Gauss's theorem to obtain the electric fields both inside and
outside each sphere. Sketch the behavior of the fields as a function of
radius for the first two spheres, and for the third with n = —2, +2.
1.4 The timeaverage potential of a neutral hydrogen atom is given by
„, e~ aT I . a.r\
®=1 — I 1 + 2)
where q is the magnitude of the electronic charge, and a 1 = a J 2. Find
the distribution of charge (both continuous and discrete) which will give
this potential and interpret your result physically.
1.5 A simple capacitor is a device formed by two insulated conductors adjacent
to each other. If equal and opposite charges are placed on the conductors,
there will be a certain difference of potential between them. The ratio of
the magnitude of the charge on one conductor to the magnitude of the
potential difference is called the capacitance (in electrostatic units it is
measured in centimeters). Using Gauss's law, calculate the capacitance of
(a) two large, fiat, conducting sheets of area A, separated by a small
distance d;
(b) two concentric conducting spheres with radii a, b (b > a) ;
(c) two concentric conducting cylinders of length L, large compared to
their radii a, b {b > a).
(d) What is the inner diameter of the outer conductor in an airfilled
coaxial cable whose center conductor is B&S#20 gauge wire and whose
capacitance is 0.5 micromicrofarad/cm ? 0.05 micromicrofarad/cm ?
1.6 Two long, cylindrical conductors of radii a x and a 2 are parallel and
separated by a distance d which is large compared with either radius.
[Probs. 1] Introduction to Electrostatics 25
Show that the capacitance per unit length is given approximately by
where a is the geometrical mean of the two radii.
Approximately what B&S gauge wire (state diameter in millimeters
as well as gauge) would be necessary to make a twowire transmission line
with a capacitance of 0.1 ju/if/cm if the separation of the wires was 0.5 cm?
1.5 cm? 5.0 cm?
1.7 (a) For the three capacitor geometries in Problem 1.5 calculate the total
electrostatic energy and express it alternatively in terms of the equal and
opposite charges Q and — Q placed on the conductors and the potential
difference between them.
(b) Sketch the energy density of the electrostatic field in each case as a
function of the appropriate linear coordinate.
1.8 Calculate the attractive force between conductors in the parallel plate
capacitor (Problem 1.5a) and the parallel cylinder capacitor (Problem 1.6)
for
(a) fixed charges on each conductor;
(b) fixed potential difference between conductors.
1.9 Prove the mean value theorem: For chargefree space the value of the
electrostatic potential at any point is equal to the average of the potential
over the surface of any sphere centered on that point.
1.10 Use Gauss's theorem to prove that at the surface of a curved charged
conductor the normal derivative of the electric field is given by
\ BE _ / 1
E~dn ~
— —\
where R ± and R 2 are the principal radii of curvature of the surface.
1.11 Prove Green's reciprocation theorem: If O is the potential due to a volume
charge density p within a volume V and a surfacecharge density a on the
surface S bounding the volume V, while €>' is the potential due to another
charge distribution p and <x', then
p$' cPx + crO' da = p'O d 3 x + a'<& da
Jv Js Jr Js
1.12 Prove Thomson's theorem : If a number of conducting surfaces are fixed in
position and a given total charge is placed on each surface, then the electro
static energy in the region bounded by the surfaces is a minimum when the
charges are placed so that every surface is an equipotential.
1.13 Prove the following theorem: If a number of conducting surfaces are
fixed in position with a given total charge on each, the introduction of an
uncharged, insulated conductor into the region bounded by the surfaces
lowers the electrostatic energy.
Boundary Value Problems
in Electrostatics: I
Many problems in electrostatics involve boundary surfaces on which
either the potential or the surfacecharge density is specified. The formal
solution of such problems was presented in Section 1.10, using the method
of Green's functions. In practical situations (or even rather idealized
approximations to practical situations) the discovery of the correct Green's
function is sometimes easy and sometimes not. Consequently a number of
approaches to electrostatic boundaryvalue problems have been developed,
some of which are only remotely connected to the Green's function
method. In this chapter we will examine two of these special techniques :
(1) the method of images, which is closely related to the use of Green's
functions; (2) expansion in orthogonal functions, an approach directly
through the differential equation and rather remote from the direct
construction of a Green's function. Other methods of attack, such as the
use of conformal mapping in twodimensional problems, will be omitted.
For a discussion of conformal mapping the interested reader may refer to
the references cited at the end of the chapter.
2.1 Method of Images
The method of images concerns itself with the problem of one or more
point charges in the presence of boundary surfaces, e.g., conductors either
grounded or held at fixed potentials. Under favorable conditions it is
possible to infer from the geometry of the situation that a small number of
suitably placed charges of appropriate magnitudes, external to the region
of interest, can simulate the required boundary conditions. These charges
26
[Sect. 2.2]
BoundaryValue Problems in Electrostatics: I
27
f =
U — $ = o
9
r
Fig. 2.1 Solution by method of
images. The original potential
problem is on the left, the
equivalentimage problem on
the right.
are called image charges, and the replacement of the actual problem with
boundaries by an enlarged region with image charges but no boundaries is
called the method of images. The image charges must be external to the
volume of interest, since their potentials must be solutions of Laplace's
equation inside the volume; the "particular integral" (i.e., solution of
Poisson's equation) is provided by the sum of the potentials of the charges
inside the volume.
A simple example is a point charge located in front of an infinite plane
conductor at zero potential, as shown in Fig. 2.1. It is clear that this is
equivalent to the problem of the original charge and an equal and opposite
charge located at the mirror image point behind the plane defined by the
position of the conductor.
2.2 Point Charge in the Presence of a Grounded Conducting
Sphere
As an illustration of the method of images we consider the problem
illustrated in Fig. 2.2 of a point charge q located at y relative to the origin
around which is centered a grounded conducting sphere of radius a. * We
seek the potential O(x) such that 0(x = a) = 0. By symmetry it is
evident that the image charge q' (assuming that only one image is needed)
will lie on the ray from the origin to the charge q. If we consider the charge
q outside the sphere, the image position y' will lie inside the sphere. The
* The term grounded is used to imply that the surface or object is held at the same
potential as the point at infinity by means of some fine conducting connector. The
connection is assumed not to disturb the potential distribution. But arbitrary amounts
of charge of either sign can flow onto the object from infinity in order to maintain its
potential at "ground" (usually taken to be zero potential). A conductor held at a fixed
potential is essentially the same situation, except that a voltage source is interposed
between the object and "ground."
28
Classical Electrodynamics
Fig. 2.2 Conducting sphere of
radius a, with charge q and image
charge q'.
potential due to the charges q and q' is :
$(x) = 72—, +
x  y x  y'
(2.1)
We now must try to choose q' and y' such that this potential vanishes at
x = a. If n is a unit vector in the direction x, and n' a unit vector in the
direction y, then
$(*) = ~ +
xn — yn'\ \xn — y'n'\
(2.2)
If x is factored out of the first term and y' out of the second, the potential
at x = a becomes:
(fr(x = a) =
q
n n
+
q'
y'
a
n n
y'
From the form of (2.3) it will be seen that the choices:
(2.3)
y _a_
a y'
make 0(a; = a) = 0, for all possible values of n • n'. Hence the magnitude
and position of the image charge are
, a , a
q = q, y = —
y y
(2.4)
[Sect. 2.2]
Boundary Value Problems in Electrostatics:
29
We note that, as the charge q is brought closer to the sphere, the image
charge grows in magnitude and moves out from the center of the sphere.
When q is just outside the surface of the sphere, the image charge is equal
and opposite in magnitude and lies just beneath the surface.
Now that the image charge has been found, we can return to the original
problem of a charge q outside a grounded conducting sphere and consider
various effects. The actual charge density induced on the surface of the
sphere can be calculated from the normal derivative of <D at the surface:
a = —
j_ao>
Andx
Ancf \y.
(;)■
('$)
(2.5)
( 1+ «_!_ 2 « cosr y
\ y 2 y i
where y is the angle between x and y. This charge density in units of
— q/4TTa 2 is shown plotted in Fig. 2.3 as a function of y for two values of
y\a. The concentration of charge in the direction of the point charge q is
evident, especially for y\a — 2. It is easy to show by direct integration
that the total induced charge on the sphere is equal to the magnitude of the
image charge, as it must according to Gauss's law.
Fig. 2.3 Surfacecharge density a
induced on the grounded sphere
of radius a due to the presence
of a point charge q located a dis
tance y away from the center of
the sphere, a is plotted in units of
—q/4TTa 2 as function of the angular
position y away from the radius
to the charge for y = 2a, 4a.
4ira 2 <r
q
30 Classical Electrodynamics
^dF=2ir(T 2 da
Fig. 2.4
The force acting on the charge q can be calculated in different ways.
One (the easiest) way is to write down immediately the force between the
charge q and the image charge q' . The distance between them is y — y' =
2/(1 — a 2 /y 2 ). Hence the attractive force, according to Coulomb's law, is:
For large separations the force is an inverse cube law, but close to the
sphere it is proportional to the inverse square of the distance away from
the surface of the sphere.
The alternative method for obtaining the force is to calculate the total
force acting on the surface of the sphere. The force on each element of
area da is lira 2 da, where a is given by (2.5), as indicated in Fig. 2.4. But
from symmetry it is clear that only the component parallel to the radius
vector from the center of the sphere to q contributes to the total force.
Hence the total force acting on the sphere (equal and opposite to the force
acting on q) is given by the integral:
1 1 + — cos 7
\ y* y J
Integration immediately yields (2.6).
The whole discussion has been based on the understanding that the
point charge q is outside the sphere. Actually, the results apply equally for
the charge q inside the sphere. The only change necessary is in the surface
charge density (2.5), where the normal derivative out of the conductor is
now radially inwards, implying a change in sign. The reader may transcribe
all the formulas, remembering that now y < a. The angular distributions
of surface charge are similar to those of Fig. 2.3, but the total induced
surface charge is evidently equal to — q, independent ofy.
[Sect. 2.3] BoundaryValue Problems in Electrostatics: I 31
2.3 Point Charge in the Presence of a Charged, Insulated,
Conducting Sphere
In the previous section we considered the problem of a point charge q
near a grounded sphere and saw that a surfacecharge density was induced
on the sphere. This charge was of total amount q' = —aqjy, and was
distributed over the surface in such a way as to be in equilibrium under all
forces acting.
If we wish to consider the problem of an insulated conducting sphere
with total charge Q in the presence of a point charge q, we can build up
the solution for the potential by linear superposition. In an operational
sense, we can imagine that we start with the grounded conducting sphere
(with its charge q' distributed over its surface). We then disconnect the
ground wire and add to the sphere an amount of charge (Q — q'). This
brings the total charge on the sphere up to Q. To find the potential we
merely note that the added charge (Q — q') will distribute itself uniformly
over the surface, since the electrostatic forces due to the point charge q are
already balanced by the charge q'. Hence the potential due to the added
charge (Q — q') will be the same as if a point charge of that magnitude
were at the origin, at least for points outside the sphere.
The potential is the superposition of (2.1) and the potential of a point
charge (Q — q') at the origin:
<D(x) =
xy[
Q + a  q
aq + y  (2.8)
a
y 2
The force acting on the charge q can be written down directly from
Coulomb's law. It is directed along the radius vector to q and has the
magnitude :
qa 3 (2y 2 — a 2 )
*"S
Q
y{y 2  a 2 ) 2 J y
(2.9)
In the limit of y > a, the force reduces to the usual Coulomb's law for two
small charged bodies. But close to the sphere the force is modified because
of the induced charge distribution on the surface of the sphere. Figure 2.5
shows the force as a function of distance for various ratios of Qjq. The
force is expressed in units of q 2 /y 2 ; positive (negative) values correspond
to a repulsion (attraction). If the sphere is charged oppositely to q, or is
32
Classical Electrodynamics
Fig. 2.5 The force on a point charge q due to an insulated, conducting sphere of radius
a carrying a total charge Q. Positive values mean a repulsion, negative an attraction.
The asymptotic dependence of the force has been divided out. Fy 2 /q 2 is plotted versus
yja for Q/q = 1, 0, 1, 3. Regardless of the value of Q, the force is always attractive
at close distances because of the induced surface charge.
uncharged, the force is attractive at all distances. Even if the charge Q is
the same sign as q, however, the force becomes attractive at very close
distances. In the limit of Q > q, the point of zero force (unstable equili
brium point) is very close to the sphere, namely, at y ~ a{\ + iVq/Q).
Note that the asymptotic value of the force is attained as soon as the charge
q is more than a few radii away from the sphere.
This example exhibits a general property which explains why an excess
of charge on the surface does not immediately leave the surface because of
mutual repulsion of the individual charges. As soon as an element of
charge is removed from the surface, the image force tends to attract it
back. If sufficient work is done, of course, charge can be removed from
the surface to infinity. The work function of a metal is in large part just
the work done against the attractive image force in order to remove an
electron from the surface.
[Sect. 2.5] BoundaryValue Problems in Electrostatics: I 33
2.4 Point Charge near a Conducting Sphere at Fixed Potential
Another problem which can be discussed easily is that of a point charge
near a conducting sphere held at a fixed potential V. The potential is the
same as for the charged sphere, except that the charge (Q — q') at the
center is replaced by a charge; (Va). This can be seen from (2.8), since at
x = a the first two terms cancel and the last term will be equal to V as
required. Thus the potential is
$(x) = — 1 — ^  1  —
x  y
a
x y
y 2
+ rf ( 2  10 >
x
The force on the charge q due to the sphere at fixed potential is
(2.11)
F2
Va qaf
y*L (y*a*fJy
For corresponding values of Va/q and Q/q this force is very similar to that
of the charged sphere, shown in Fig. 2.5, although the approach to
the asymptotic value (Vaqfy 2 ) is more gradual. For Va > q, the unstable
equilibrium point has the equivalent location y ~ a(\ + ^Vq/Va).
2.5 Conducting Sphere in a Uniform Electric Field by Method
of Images
As a final example of the method of images we consider a conducting
sphere of radius a in a uniform electric field E . A uniform field can be
thought of as being produced by appropriate positive and negative charges
at infinity. For example, if there are two charges ± Q, located at positions
z = ^R, as shown in Fig. 2.6a, then in a region near the origin whose
dimensions are very small compared to R there is an approximately
constant electric field E ^ 2 Q/R 2 parallel to the z axis. In the limit as
R, Q^co, with Q/R 2 constant, this approximation becomes exact.
If now a conducting sphere of radius a is placed at the origin, the
potential will be that due to the charges ±Q at =fR and their images
=FQalRatz= Ta 2 /R:
Q Q
o =
( r 2 + #2 + lr R cos Of 4 (r 2 + R 2  2rR cos 0) H
^Q + ^2 (2 12)
R ( r z + ± + ll2L cos e) R (r 2 + °L  ^ cos 6
\ R 2 R J \ R 2 R
34
Classical Electrodynamics
. — ■■ —
P
~~~~~—  __«
z=R
z = R
(a)
z=R
(b)
Fig. 2.6 Conducting sphere in a uniform electric field by the method of images.
where <J> has been expressed in terms of the spherical coordinates of the
observation point. In the first two terms R is much larger than r by
assumption. Hence we can expand the radicals after factoring out R 2 .
Similarly, in the third and fourth terms, we can factor out r 2 and then
expand. The result is :
<D =
2Q a , 2Qa 3
^ r cos V \ *r — cos
R 2 R 2 r 2
+
(2.13)
where the omitted terms vanish in the limit R —* oo. In that limit 2Q/R 2
becomes the applied uniform field, so that the potential is
(2.14)
O = —EJr — —\ cos 6
The first term (— E z) is, of course, just the potential of a uniform field E
which could have been written down directly instead of the first two terms
in (2.12). The second is the potential due to the induced surface charge
density or, equivalently, the image charges. Note that the image charges
form a dipole of strength D = Qa/R x 2a 2 /R = E a s . The induced
surfacecharge density is
477 dr
= — E cos
4tt
(2.15)
[Sect. 2.6] BoundaryValue Problems in Electrostatics: I 35
We note that the surface integral of this charge density vanishes, so that
there is no difference between a grounded and an insulated sphere.
2.6 Method of Inversion
The method of images for a sphere and related topics discussed in the
previous sections suggest that there is some sort of equivalence of solutions
of potential problems under the reciprocal radius transformation,
a 2
r>r' =  (2.16)
r
This equivalence forms the basis of the method of inversion, and trans
formation (2.16) is called inversion in a sphere. The radius of the sphere is
called the radius of inversion, and the center of the sphere, the center of
inversion. The mathematical equivalence is contained in the following
theorem :
Let 0(r, 6, <f>) be the potential due to a set of point charges q t at the
points (r„ f , </> t ). Then the potential
O'(r,0 ; «£) = ^O^\0,<^ (2.17)
is the potential due to charges,
q/ =  qi (2.18)
located at the points (a 2 /r 4 , d t , </>,).
The proof of the theorem is as follows. The potential <b{r, d, <f>) can be
written as
*=2
<it
Vr 2 + r, 2  2rr t
cosy*
where y 4 is the angle between the radius vectors x and x t . Under trans
formation (2.16) the angles remain unchanged. Consequently the new
potential <J>' is
 , 2 2a 2
— + r/ r t cos y t
r r
36
Classical Electrodynamics
$ . p
1.2q
$ ' p
6q
Fig. 2.7
By factoring (r//r 2 ) out of the square root, this can be written
(?)
O
'(r, 6, <f>) = ^
r + —  2r — cos y,
This proves the theorem.
Figure 2.7 shows a simple configuration of charges before and after
inversion. The potential O' at the point P due to the inverted distribution
of charge is related by (2.17) to the original potential O at the point P' in
the figure.
The inversion theorem has been stated and proved with discrete charges.
It is left as an exercise for the reader to show that, if the potential O
satisfies Poisson's equation,
V 2 0> = Anp
the new potential <D' (2.17) also satisfies Poisson's equation,
V 2 <D'(r, d, $) = 47rp'(r, Q, cf>)
where the new charge density is given by
P '(r,d,cf>)=^j P (£,e,+)
(2.19)
(2.20)
The connection between this transformation law for charge densities and
the law (2.18) for point charges can be established by considering the
charge density as a sum of delta functions :
p(x) = 2,q i d(xx i )
[Sect. 2.6] BoundaryValue Problems in Electrostatics: I 37
In terms of spherical coordinates centered at the center of inversion the
charge density can be written
P {r, B, <f>) = y qi m  &d  2 Kr ~ rd
r r i
where d(Q, — Q. t ) is the angular delta function whose integral over solid
angle gives unity, and d(r — r t ) is the radial delta function.* Under
inversion the angular factor is unchanged. Consequently we have
p{j > 6 ><i>) = 2** Q  °*> f 2 6 {^  r >)
The radial delta function can be transformed according to rule 5 at the
end of Section 1.2 as
Ati
Then
P (£,o,t) = 2^°°^
6 /^,2\2
and the inverted charge density (2.20) becomes
P '(r, d, <f>) = ^ 2* (fj ^ x ~ x <') = 2«'^ x ~ x/)
i * i
where x/ = (a 2 /r,, 0, <£) and q = (a/r t )q i: , as required by (2.18).
With the transformation laws for charges and volumecharge densities
given by (2.18) and (2.20), it will not come as a great surprise that the
transformation of surfacecharge densities is according to
o'(r,d,$)= ^0(^,6,^ (2.21)
Before treating any examples of inversion there are one or two physical
and geometrical points which need discussion. First, in regard to the
physical points, if the original potential problem is one where there are
conducting surfaces at fixed potentials, the inverted problem will not in
general involve the inversions of those surfaces held at fixed potentials.
This is evident from (2.17), where the factor a\r shows that even if <I> is
constant on the original surface the potential O' on the inverted surface is
* The factor r { ~ 2 multiplying the radial delta function is present to cancel out the r 2
which appears in the volume element d 3 x = r 2 dr dCl.
38
Classical Electrodynamics
Fig. 2.8 Geometry of inversion.
Center of inversion is at 0. Radius
of inversion is a. The inversion of
the surface 5 is the surface S', and
vice versa.
not. The only exception occurs when <J> vanishes on some surface. Then
O' also vanishes on the inverted surface.
One might think that, since ^> is arbitrary to the extent of an additive
constant, we could make any surface in the original problem have zero
potential and so also be at zero potential in the inverted problem. This
brings us to the second physical point. The inverted potentials corre
sponding to two potential problems differing only by an added constant
potential <J> represent physically different charge configurations, namely,
charge distributions which differ by a point charge a<& located at the center
of inversion. This can be seen from (2.17), where a constant term <D in <1>
is transformed into (aO /r). Consequently care must be taken in applying
the method of inversion to remember that the mapping of the point at
infinity into the origin may introduce point charges there. If these are not
wanted, they must be separately removed by suitable linear superposition.
The geometrical considerations involve only some elementary points
which can be proved very simply. The notation is shown in Fig. 2.8. Let
O be the center of inversion, and a the radius of inversion. The inter
section of the sphere of inversion and the plane of the paper is shown as
the dotted circle. A surface S intersects the page with the curve AB. The
inverted surface S', obtained by transformation (2.16), intersects the page
in the curve A'B'. The following facts are stated without proof:
(a) Angles of intersection are not altered by inversion.
(b) An element of area da on the surface S is related to an element of
area da' on the inverted surface S' by da/da' = r 2 /r' 2 .
(c) The inverse of a sphere is always another sphere [perhaps of infinite
radius; see (d)].
(d) The inverse of any plane is a sphere which passes through the center
of inversion, and conversely.
Figure 2.9 illustrates the possibilities involved in (c) and (d) when the
center of inversion lies outside, on the surface of, or inside the sphere.
[Sect. 2.6]
Boundary Value Problems in Electrostatics: I
39
As a very simple example of the solution of a potential problem by
inversion we consider an isolated conducting sphere of radius R with a
total charge Q on it. The potential has the constant value Q/R inside the
sphere and falls off inversely with distance away from the center for points
outside the sphere. By a suitable choice of center of inversion and
associated parameters we can obtain the potential due to a point charge q
a distance i/away from an infinite, grounded, conducting plane. Evidently,
if the center of inversion O is chosen to lie on the surface of the sphere of
radius R, the sphere will invert into a plane. This geometric situation is
shown in Fig. 2.10. Furthermore, if we choose the arbitrary additive
constant potential % to have the value — QjR, the sphere and its inversion,
the plane, will be at zero potential, while a point charge —aQ/R will appear
at the center of inversion. In order that we end up with a point charge q a
distance d away from the plane it is necessary to choose the radius of
inversion to be a = {2Rd)' A and the initial charge, Q = —(Rj2df*q. The
surfacecharge density induced on the plane can be found easily from (2.21).
Since the charge density on the sphere is uniform over its surface, the
induced charge density on the plane varies inversely as the cube of the
distance away from the origin (as can be verified from the image solution;
see Problem 2.1).
If the center of inversion is chosen to lie outside the isolated uniformly
charged sphere, it is clear from Fig. 2.9 that the inverted problem can be
Fig. 2.9 Various possibilities for the inversion of a sphere. If the center of inversion O
lies on the surface 5 of the sphere, the inverted surface S' is a plane; otherwise it is
another sphere. The sphere of inversion is shown dotted.
40
Classical Electrodynamics
■ — f— 
2R
■*!
Fig. 2.10 Potential due to isolated, charged,
conducting sphere of radius R is inverted to give
the potential of a point charge a distance d
away from an infinite, flat, conducting surface.
made that of a point charge near a grounded conducting sphere, handled
by images in Section 2.2. The explicit verification of this is left to Problem
2.9.
A very interesting use of inversion was made by Lord Kelvin in 1 847.
He calculated the charge densities on the inner and outer surfaces of a thin,
charged, conducting bowl made from a sphere with a cap cut out of it. The
potential distribution which he inverted was that of a thin, flat, charged,
circular disc (the charged disc is discussed in Section 3.12). As the shape
of the bowl is varied from a shallow watch glasslike shape to an almost
closed sphere, the charge densities go from those of the disc to those of a
closed sphere, in the one limit being almost the same inside and out, but
concentrated at the edges of the bowl, and in the other limit being almost
zero on the inner surface and uniform over the outer surface. Numerical
values are given in Kelvin's collected papers, p. 186, and in Jeans, pp.
250251.
2.7 Green's Function for the Sphere; General Solution
for the Potential
In preceding sections the problem of a conducting sphere in the presence
of a point charge has been discussed by the method of images. As was
mentioned in Section 1.10, the potential due to a unit charge and its image
(or images), chosen to satisfy homogeneous boundary conditions, is just
[Sect. 2.7]
BoundaryValue Problems in Electrostatics:
41
the Green's function (1.43 or 1.45) appropriate for Dirichlet or Neumann
boundary conditions. Tn G^x, x') the variable x' refers to the location P'
of the unit charge, while the variable x is the point P at which the potential
is being evaluated. These coordinates and the sphere are shown in Fig.
2. 1 1 . For Dirichlet boundary conditions on the sphere of radius a the
potential due to a unit charge; and its image is given by (2.1) with q = 1
and relations (2.4). Transforming variables appropriately, we obtain the
Green's function:
1
G(x, x') =
In terms of spherical coordinates this can be written :
1 1
G(x, x') =
(2.22)
(*■ + **  2^ cos yT l^ + a *_ lxx , cos y J
(2.23)
where y is the angle between x and x'. The symmetry in the variables x
and x' is obvious in the form (2.23), as is the condition that G = if either x
or x' is on the surface of the sphere.
Fig. 2.11
42
Classical Electrodynamics
For solution (1.44) of Poisson's equation we need not only G, but also
dG/dri. Remembering that n' is the unit normal outwards from the
volume of interest, i.e., inwards along x' toward the origin, we have
dG
dn'
(x 2  a 2 )
a{x 2 + a 2 — lax cos y)'
(2.24)
[Note that this is essentially the induced surfacecharge density (2.5).]
Hence the solution of Laplace's equation outside a sphere with the potential
specified on its surface is, according to (1.44),
0(x) = f U(a, B', <f>'
a(x 2 — a 2 )
(x 2 + a 2 — 2a x cos yj
7 dV (2.25)
where dO.' is the element of solid angle at the point (a, 6', </>') and cos y =
cos 6 cos 0' + sin sin 0' cos (<£ — <£')• For the interior problem, the
normal derivative is radially outwards, so that the sign of dG/dn' is opposite
to (2.24). This is equivalent to replacing the factor (x 2 — a 2 ) by (a 2 — x 2 )
in (2.25). For a problem with a charge distribution, we must add to (2.25)
the appropriate integral in (1.44), with the Green's function (2.23).
2.8 Conducting Sphere with Hemispheres at Different Potentials
As an example of general solution for the potential outside a sphere
with prescribed values of potential on its surface, we consider the con
ducting sphere of radius a made up of two hemispheres separated by a small
insulating ring. The hemispheres are kept at different potentials. It will
suffice to consider the potentials as ± V, since arbitrary potentials can be
handled by superposition of the solution for a sphere at fixed potential
over its whole surface. The insulating ring lies in the z = plane, as
shown in Fig. 2.12, with the upper (lower) hemisphere at potential +V
(V).
Fig. 2.12
[Sect. 2.8] BoundaryValue Problems in Electrostatics: I 43
From (2.25) the solution for 0(#, 6, <f>) is given by the integral:
<D(*, 6,<f>) = T PVf \ \/(cos 0')  f ° d(cos 0')} , 2 "f~ a2) ^
47rJo Uo Ji ) (a + a — lax cos y)^
(2.26)
By a suitable change of variables in the second integral (0' — >■ 7r — 0',
<f>' + <f>' + 77), this can be cast in the form :
*(*,M) =
Fa(z 2
Jo Jo
d(cos d')[(a 2 + x 2 — laxcos y)~ 3/i
 (a 2 + x 2 + lax cos y)  ^] (2.27)
Because of the complicated dependence of cos y on the angles (0', </>') and
(6, (f>), equation (2.27) cannot in general be integrated in closed form.
As a special case we consider the potential on the positive z axis. Then
cos y = cos 6' since = 0. The integration is elementary, and the
potential can be shown to be
(z 2 «V
<D(z) = V
1 
sVz 2 + a 2  1
(2.28)
At 2 = a, this reduces to <D == F as required, while at large distances it
goes asymptotically as ~ 3 Va 2 /2z 2 .
In the absence of a closed expression for the integrals in (2.27), we can
expand the denominator in power series and integrate term by term.
Factoring out (a 2 + x 2 ) from each denominator, we obtain
4tt(x 2 +
2\ /*2jt /*1
4U ^'U(cos0')[(l2acos r ) %
ar)* Jo Jo
(l + 2acosy) M ] (2.29)
where a = axj(a 2 + a 2 ). We observe that in the expansion of the radicals
only odd powers of a cos y will appear:
[(1  2a cos y)~ 3A — (1 + 2a cos y)~ 3A ] = 6a cos 7 + 35a 3 cos 3 y +
(2.30)
It is now necessary to integrate odd powers of cos y over d<f>' d(cos 0') :
dcf>' d(cos 0') cos y = 7T cos
Jo Jo
Jo Jo
d(cos 0') cos 3 y =  cos 0(3  cos 2 0)
4
(2.31)
44 Classical Electrodynamics
If (2.30) and (2.31) are inserted into (2.29), the potential becomes
°(*' *' *> = ^j l J cos Q
2x £ \ (x* + ay
(2.32)
1+ ** f3cos 2 6) +
. 24(a 2 + x 2 ? K
We note that only odd powers of cos 6 appear, as required by the symmetry
of the problem. If the expansion parameter is (a 2 /x 2 ), rather than a 2 , the
series takes on the form :
o(M, <t>) = 3Va2
2x 2
n 7 « 2 P 3 fl 3 A ,
COS0 cos 3 COS0) +
12a; 2 \2 2 /
(2.33)
For large values of xja this expansion converges rapidly and so is a useful
representation for the potential. Even for xja = 5, the second term in the
series is only of the order of 2 per cent. It is easily verified that, for
cos 0=1, expression (2.33) agrees with the expansion of (2.28) for the
potential on the axis. [The particular choice of angular factors in (2.33) is
dictated by the definitions of the Legendre polynomials. The two factors
are, in fact, P x {cos 6) and ^(cos 6), and the expansion of the potential is
one in Legendre polynomials of odd order. We shall establish this in a
systematic fashion in Section 3.3.]
2.9 Orthogonal Functions and Expansions
The representation of solutions of potential problems (or any mathe
matical physics problem) by expansions in orthogonal functions forms a
powerful technique that can be used in a large class of problems. The
particular orthogonal set chosen depends on the symmetries or near
symmetries involved. To recall the general properties of orthogonal
functions and expansions in terms of them, we consider an interval (a, b)
in a variable  with a set of real or complex functions U n (£), n = 1,2,...,
orthogonal on the interval (a, b). The orthogonality condition on the
functions UJJ) is expressed by
V n *(0tf m (0 & = 0, m^n (2.34)
r
lfn = m, the integral is finite. We assume that the functions are normal
ized so that the integral is unity. Then the functions are said to be
orthonormal, and they satisfy
f
*> a
U n *(OU m (£)d£ = d nm (2.35)
[Sect. 2.9] BoundaryValue Problems in Electrostatics: I 45
An arbitrary function /(I), square integrable on the interval {a, b), can
be expanded in a series of the orthonormal functions UJJ). If the number
of terms in the series is finite (say N),
f(£)^la n UM (2.36)
n = l
then we can ask for the "best" choice of coefficients a n so that we get the
"best" representation of the function /(I). If "best" is defined as mini
mizing the mean square error M N :
M N = P \m  I a n UM
"a I n=l
dt (2.37)
it is easy to show that the coefficients are given by
U n *(S)f(£) di (2.38)
where the orthonormality condition (2.35) has been used. This is the
standard result for the coefficients in an orthonormal function expansion.
If the number of terms N in series (2.36) is taken larger and larger, we
intuitively expect that our series representation of /(£) is "better" and
"better." Our intuition will be correct provided the set of orthonormal
functions is complete, completeness being defined by the requirement that
there exist a finite number N such that for N > N the mean square error
M N can be made smaller than any arbitrarily small positive quantity. Then
the series representation
oo
Ia n U n (i)=f(0 (2.39)
« = :l
with a n given by (2.38) is said to converge in the mean to /(£). Physicists
generally leave the difficult job of proving completeness of a given set of
functions to the mathematicians. All orthonormal sets of functions
normally occurring in mathematical physics have been proved to be
complete.
Series (2.39) can be rewritten with the explicit form (2.38) for the
coefficients a n :
/(£) = [ { I U n *(i')U n (i)} fin di' (2.40)
Since this represents any function /(!) on the interval (a, b), it is clear that
the sum of bilinear terms U n *(i')U n {£) must exist only in the neighborhood
of I' = . In fact, it must be true that
2 U n *(t')UM = <5(£'  ) (2.41)
46 Classical Electrodynamics
This is the socalled completeness or closure relation. It is analogous to the
orthonormality condition (2.35), except that the roles of the continuous
variable £ and the discrete index n have been interchanged.
The most famous orthogonal functions are the sines and cosines, an
expansion in terms of them being a Fourier series. If the interval in x is
(—a/2, a/2), the orthonormal functions are
2 . (iTrmxX fl (iTrmxX
sin I 1, /cos I I
a \ a / N a \ a 1
where m is an integer. The series equivalent to (2.39) is customarily
written in the form: .
TO = 1
where
, 2 f a/2 ,, , (2irmx\ ,
A m =  f(x) cos dx
a J a/2 \ a I
(2.43)
B m =  f(x) sin I rfa;
a ^o/2 \ a /
If the interval spanned by the orthonormal set has more than one
dimension, formulas (2.34)(2.39) have obvious generalizations. Suppose
that the space is two dimensional, and that the variable I ranges over the
interval (a, b) while the variable r\ has the interval (c, d). The orthonormal
functions in each dimension are U n (tj) and V m (rj). Then the expansion of
an arbitrary function /(I, rj) is
/(*, V)=1I a n JJM)VJri) (2.44)
n m
where
a nm = f " d£ T driU n *(i)V m \rj)f(S, rj) (2.45)
If the interval (a, b) becomes infinite, the set of orthogonal functions
U n (£) may become a continuum of functions, rather than a denumerable
set. Then the Kronecker delta symbol in (2.35) becomes a Dirac delta
function. An important example is the Fourier integral. Start with the
orthonormal set of complex exponentials,
UJx) = 4= e i(2 " mx/a) (2.46)
m = 0, ±1, ±2, . . . , on the interval (—all, a/2), with the expansion:
1 °o
f(x) = = y A m e i{  2lTmxla) (2.47)
Ja *—'
V m= — oo
[Sect. 2.10] BoundaryValue Problems in Electrostatics: I 47
where
Am = _L j ° /2 e n»™*Mf( x ')dx' (2.48)
m ^/a J a/2
Then let the interval become infinite {a > oo), at the same time trans
forming
27rm
k
^ Joo 27rJc
dk
A(k)
The resulting expansion, equivalent to (2.47), is
1
/(«)=! A{ky k *dk
.. / Z7T J  oo
where
A(k) = L  * e~ ikx f(x) dx
(2.49)
^277
The orthogonality condition is
— f°° e^ k  k ' }x dx = d(k  W)
2.7T v — oo
while the completeness relation is
J_ * ^JK**') ^^ _ fi( x _ 3.')
27T Joo
These last integrals serve as convenient representations of a delta function.
We note in (2.50)(2.53) the complete equivalence of the two continuous
variables x and k.
(2.50)
(2.51)
(2.52)
(2.53)
2.10 Separation of Variables; Laplace's Equation
in Rectangular Coordinates
The partial differential equations of mathematical physics are often
solved conveniently by a method called separation of variables. In the
process, one often generates orthogonal sets of functions which are useful
in their own right. Equations involving the threedimensional Laplacian
operator are known to be separable in eleven different coordinate systems
48 Classical Electrodynamics
(see Morse and Feshbach, pp. 509, 655). We will discuss only three of these
in any detail— rectangular, spherical, and cylindrical— and will begin with
the simplest, rectangular coordinates.
Laplace's equation in rectangular coordinates is
a 2 o , a 2 o , a 2 o> n
A solution of this partial differential equation can be found in terms of
three ordinary differential equations, all of the same form, by the assumption
that the potential can be represented by a product of three functions, one
for each coordinate:
<!>(*, y, z) = X(x) Y(y)Z(z) (2.55)
Substitution into (2.54) and division of the result by (2.55) yields
X(x) dx 2 + Y(y) dy 2 Z(z) dz 2 (2>56)
where total derivatives have replaced partial derivatives, since each term
involves a function of one variable only. If (2.56) is to hold for arbitrary
values of the independent coordinates, each of the three terms must be
separately constant:
}_<£X = _ 2
X dx 2 ~
Ydy*
1 d 2 Z
 = y 2,
Zdz* r
.2
(2.57)
where a 2 + /5 2 = y 2
If we arbitrarily choose a 2 and /5 2 to be positive, then the solutions of the
three or dinary d ifferential equations (2.57) are exp(±/aa;); exp(±/ / S«/) >
exp (±Va 2 + £ 2 z). The potential (2.55) can thus be built up from the
product solutions :
At this stage a and @ are completely arbitrary. Consequently (2.58), by
linear superposition, represents a very large class of solutions to Laplace's
equation.
To determine a and /S it is necessary to impose specific boundary
conditions on the potential. As an example, consider a rectangular box,
located as shown in Fig. 2.13, with dimensions (a, b, c) in the (x, y, z)
[Sect. 2.10] BoundaryValue Problems in Electrostatics: I
z
49
f = 0
Fig. 2.13 Hollow, rectangular
box with five sides at zero
potential, while the sixth (z = c)
has the specified potential <D =
V(x, y).
f = V(xy)
4 4
t.
y = b
f> =
* =
directions. All surfaces of the box are kept at zero potential, except the
surface z = c, which is at a potential V(x, y). It is required to find the
potential everywhere inside the box. Starting with the requirement that
<E> = for x = 0, y = 0, z = 0, it is easy to see that the required forms of
X, Y,Z are
X = sin ax
Y = sin /fy
Z = sinh (Va 2 + 0*z)
(2.59)
In order that O = at x = a and y = b, it is necessary that cm = mr and
(ib = rmr. With the definitions,
mr
OL„ =
mv
b
Vnm =7r
" 2 b 2
(2.60)
we can write the partial potential $ wm ; satisfying all the boundary
conditions except one,
<D nTO = sin (a w ar) sin ($„&) sinh (y nm z)
(2.61)
The potential can be expanded in terms of these 3>„ m with initially arbitrary
coefficients (to be chosen to satisfy the final boundary condition):
oo
0(ic, y,z)= 2 A nm sin (a w *) sin (fi m y) sinh (y nm z) (2.62)
50 Classical Electrodynamics
There remains only the boundary condition <X> = V(x, y)atz = c:
oo
V(x, y) = 2 A nm sin (pt n x) sin (p m y) sinh (y WTO c) (2.63)
n,m = l
This is just a double Fourier series for the function V(x, y). Consequently
the coefficients A nm are given by:
4 f a P
^»™ = ,. w : dx dyV(x, y) sin (a B a;) sin (0 m y) (2.64)
flfc sinh (y nTO c) Jo Jo
If the rectangular box has potentials different from zero on all six sides,
the required solution for the potential inside the box can be obtained by a
linear superposition of six solutions, one for each side, equivalent to (2.62)
and (2.64). The problem of the solution of Poisson's equation, i.e., the
potential inside the box with a charge distribution inside, as well as
prescribed boundary conditions on the surface, requires the construction of
the appropriate Green's function, according to (1.43) and (1.44). Discus
sion of this topic will be deferred until we have treated Laplace's equation
in spherical and cylindrical coordinates. For the moment, we merely note
that solution (2.62) and (2.64) is equivalent to the surface integral in the
Green's function solution (1 .44).
REFERENCES AND SUGGESTED READING
Images and inversion are treated in many books ; among the better or more extensive
discussions are those by
Jeans, Chapter VIII,
Maxwell, Vol. 1, Chapter XI,
Smythe, Chapters IV and V.
A truly encyclopedic source of examples with numerous diagrams is the book by
Durand, especially Chapters III and IV.
Durand discusses inversion on pp. 107114.
Conformal mapping techniques for the solution of twodimensional potential problems
are discussed by
Durand, Chapter X,
Jeans, Chapter VIII, Sections 306337,
Maxwell, Vol. 1, Chapter XII,
Smythe, Chapter IV, Sections 4.09^1.29.
There are, in addition, many engineering books devoted to the subject, e.g.,
Rothe, Ollendorff, and Polhausen.
Elementary, but clear, discussions of the mathematical theory of Fourier series and
integrals, and orthogonal expansions, can be found in
Churchill,
Hildebrand, Chapter 5.
A somewhat oldfashioned treatment of Fourier series and integrals, but with many
examples and problems, is given by
Byerly.
[Probs. 2] BoundaryValue Problems in Electrostatics:
51
PROBLEMS
2.1 A point charge q is brought to a position a distance d away from an infinite
plane conductor held at zero potential. Using the method of images, find:
(a) the surfacecharge density induced on the plane, and plot it;
(b) the force between the plane and the charge by using Coulomb's law
for the force between the charge and its image;
(c) the total force acting on the plane by integrating 2no 2  over the whole
plane;
(d) the work necessary to remove the charge q from its position to
infinity ;
(e) the potential energy between the charge q and its image [compare the
answer to id) and discuss].
(/) Find answer (d) in electron volts for an electron originally one
angstrom from the surface.
2.2 Using the method of images, discuss the problem of a point charge q
inside a hollow, grounded, conducting sphere of inner radius a. Find
(a) the potential inside the sphere;
(b) the induced surfacecharge density;
(c) the magnitude and direction of the force acting on q.
Is there any change in the solution if the sphere is kept at a fixed potential
VI If the sphere has a total charge Q on it?
2.3 Two infinite, grounded, conducting planes are located at x = a/2 and
x = a/2. A point charge q is placed between the planes at the point
(x', y', z'), where —(a/2) < x' < (a/2).
(a) Find the location and magnitude of all the image charges needed to
satisfy the boundary conditions on the potential, and write down the
Green's function G(x, x')
(b) If the charge a is at (*', 0, 0), find the surfacecharge densities
induced on each conducting plane and show that the sum of induced
charge on the two planes is —q.
2.4 Consider a potential problem in the halfspace defined by z > 0, with
Dirichlet boundary conditions on the plane z = (and at infinity).
(a) Write down the appropriate Green's function G(x, x').
(b) If the potential on the plane z = is specified to be $ = V inside a
circle of radius a centered at the origin, and <D = outside that circle, find
an integral expression for the potential at the point P specified in terms of
cylindrical coordinates (p, cj>, z).
(c) Show that, along the axis of the circle (p = 0), the potential is given by
$■ = V
\ Va 2 + z 2 )
{d) Show that at large distances (p 2 + z 2 > a 2 ) the potential can be
expanded in a power series in (p 2 + z 2 ) 1 , and that the leading terms are
2 ( p 2 + z 2fA
_ 3a 2 5(3p 2 a 2 + a 4 )
. 4(p 2 + z 2 ) + 8(p 2 + z 2 ) 2
2.5
2.6
52 Classical Electrodynamics
Verify that the results of (c) and (d) are consistent with each other in their
common range of validity.
An insulated spherical, conducting shell of radius a is in a uniform electric
i?i £ ,i i 16 ^ 6r / is CUt int ° tw ° hemis pheres by a plane perpendicular
to the held, find the force required to prevent the hemispheres from separa
te) if the shell is uncharged;
(b) if the total charge on the shell is Q.
A large parallel plate capacitor is made up of two plane conducting sheets
one of which has a small hemispherical boss of radius a on its inner surface
The conductor with the boss is kept at zero potential, and the other
conductor is at a potential such that far from the boss the electric field
between the plates is E .
(a) Calculate the surfacecharge densities at an arbitrary point dn the
plane and on the boss, and sketch their behavior as a function of distance
(or angle).
(b) Show that the total charge on the boss has the magnitude 3£ a 2 /4
(c) If, instead of the other conducting sheet at a different potential a
point charge q is placed directly above the hemispherical boss at a distance
d from its center, show that the charge induced on the boss is
q — —q 1 =====
L dVd* + a 2 _
2.7 A line charge with linear charge density t is placed parallel to, and a distance
R away from, the axis of a conducting cylinder of radius b held at fixed
voltage such that the potential vanishes at infinity. Find
(a) the magnitude and position of the image charge(s);
(b) the potential at any point (expressed in polar coordinates with the
line from the cylinder axis to the line charge as the x axis), including the
asymptotic form far from the cylinder;
(c) the induced surfacecharge density, and plot it as a function of angle
for Rib = 2, 4 in units of t/2ttZ>; b
(d) the force on the charge.
2.8 (a) Find the Green's function for the twodimensional potential problem
with the potential specified on the surface of a cylinder of radius b, and
show that the solution inside the cylinder is given by Poisson's integral:
(b) Two halves of a long conducting cylinder of radius b are separated
by a small gap, and are kept at different potentials V 1 and V 2 . Show that
the potential inside is given by
^t m v i + V 2 V,  V 2
<£(r, 0) = J— — ? + _1 ? tan"
( 2br r\
where 6 is measured from a plane perpendicular to the plane through the
g a P
(c) Calculate the surfacecharge density on each half of the cylinder.
(d) What modification is necessary in (a) if the potential is desired in the
region of space bounded by the cylinder and infinity?
[Probs. 2] BoundaryValue Problems in Electrostatics: I 53
2.9 (a) An isolated conducting sphere is raised to a potential V. Write down
the (trivial) solution for the electrostatic potential everywhere in space.
(b) Apply the inversion theorem, choosing the center of inversion
outside the conducting sphere. Show explicitly that the solution obtained
for the potential is that of a grounded sphere in the presence of a point charge
of magnitude  VR, where R is the inversion radius.
(c) What is the physical situation described by the inverted solution if
the center of inversion%es inside the conducting sphere?
2.10 Knowing that the capacitance of a thin, flat, circular, conducting disc of
radius a is {Ij^a and that; the surfacecharge density on an isolated disc
raised to a given potential is proportional to (a 2  r 2 )~ 1/2 , where r is the
distance from the center of the disc,
(a) show that by inversion the potential can be found for the problem
of an infinite, grounded, conducting plane with a circular hole in it and a
point charge lying anywhere in the opening;
(b) show that, for a unit point charge at the center of the opening, the
induced charge density on the plane is
<*(.r., 6,<t>) = , a
^r 2 Vr 2  a 2
(c) show that (a) and (b) are a special case of the general problem,
obtained by inversion of the disc, of a grounded, conducting, spherical
bowl under the influence of a point charge located on the cap which is the
complement of the bowl.
2.11 A hollow cube has conducting walls defined by six planes x = y = z = 0,
and x = y = z = a. The walls z = and z = a are held at a constant
potential V. The other four sides are at zero potential.
(a) Find the potential fl>(x, y, z) at any point inside the cube.
(b) Evaluate the potential at the center of the cube numerically, accurate
to three significant figures. How many terms in the series is it necessary to
keep in order to attain this accuracy? Compare your numerical result
with the average value of the potential on the walls.
(c) Find the surfacecharge density on the surface z = a.
Boundary Value Problems
in Electrostatics: II
In this chapter the discussion of boundaryvalue problems is con
tinued. Spherical and cylindrical geometries are first considered, and
solutions of Laplace's equation are represented by expansions in series of
the appropriate orthonormal functions. Only an outline is given of the
solution of the various ordinary differential equations obtained from
Laplace's equation by separation of variables, but an adequate summary of
the properties of the different functions is presented.
The problem of construction of Green's functions in terms of ortho
normal functions arises naturally in the attempt to solve Poisson's equation
in the various geometries. Explicit examples of Green's functions are
obtained and applied to specific problems, and the equivalence of the
various approaches to potential problems is discussed.
3.1 Laplace's Equation in Spherical Coordinates
In spherical coordinates (r, 6, (j>), shown in Fig. 3.1, Laplace's equation
can be written in the form :
(KP) + — sin — l+— ^ = (3.1)
rdr 2 r*smddd\ 30/ r 2 sin 2 6 d<j> 2
If a product form for the potential is assumed, then it can be written:
<D=^P(0)2(<£) (3.2)
r
54
[Sect. 3.1] BoundaryValue Problems in Electrostatics: II
55
Fig. 3.1
When this is substituted into (3.1), there results the equation
UQ d
* dr 2 r 2 sin dO
sin — I
\ del
_UP_*Q =
r 2 sin 2 6 d<j> 2
If we multiply by r 2 sin 2 d/UPQ, we obtain
1
r sin
1 d 2 U
iU dr 2 r" sin OP dO
d i . a dP
sin u —
dd
+ ^^ = (3.3)
The <f> dependence of the equation has now been isolated in the last term.
Consequently that term must be a constant which we call (— m 2 ):
This has solutions
1 d 2 Q 2
= — m 2
Qd</> 2
Q = e
±im</>
(3.4)
(3.5)
In order that Q be single valued, m must be an integer. By similar con
siderations we find separate equations for P(d) and U(r):
1
sin0d0\
d ( . dP\
— I sin u — )
id\ dd)
/(/ + i)
m
P =
*E  l(l + 1} u = o
dr 2 r 2
(3.6)
(3.7)
where /(/ + 1) is another real constant.
From the form of the radial equation it is apparent that a single power
of r (rather than a power series) will satisfy it. The solution is found to be :
U = Ar^ 1 + Br~ l (3.8)
but / is as yet undetermined.
56 Classical Electrodynamics
3.2 Legendre Equation and Legendre Polynomials
The 6 equation for P(0) is customarily expressed in terms of x = cos 6,
instead of 6 itself. Then it takes the form:
d_
dx
(«*>t) + ( k, + 1) [=*) p  (3  9)
This equation is called the generalized Legendre equation, and its solutions
are the associated Legendre functions. Before considering (3.9) we will
outline the solution by power series of the ordinary Legendre differential
equation with m 2 = :
j ((1  x 2 ) f) + /(/ + 1)P = (3.10)
The desired solution should be single valued, finite, and continuous on the
interval — 1 < x < 1 in order that it represents a physical potential. The
solution will be assumed to be represented by a power series of the form :
P(x) = x*2 aj x j (3.11)
i =
where a is a parameter to be determined. When this is substituted into
(3.10), there results the series:
{(«+/X«+jiK*" +i " 2
 [(a + ;)(<* + J + 1) ~ /(/ + l»i*" + '} = (312)
In this expansion the coefficient of each power of a; must vanish separately.
Fory = 0, 1 we find that
if a =£ 0, then a(a — 1) = ]
if a x ^ 0, then a(a + 1) = J
while for a general j value
■(a+jXa+j + 1) Z(Z+1)
a i + 2 —
(3.13)
(3.14)
(a + j + l)(a +7 + 2)
A moment's thought shows that the two relations (3.13) are equivalent and
that it is sufficient to choose either a or a x different from zero, but not both.
Making the former choice, we have a = or a = 1. From (3.14) we see
that the power series has only even powers of x(<x. = 0) or only odd
powers of x(<x. = 1).
[Sect. 3.2] BoundaryValue Problems in Electrostatics: II 57
For either of the series a = or a = 1 it is possible to prove the
following properties:
(a) the series converges for x % < 1, regardless of the value of /;
(b) the series diverges at x ■— ±1, unless it terminates.
Since we want a solution that is finite at x = ± 1, as well as for x % < 1, we
demand that the series terminate. Since a andy are positive integers or
zero, the recurrence relation (3.14) will terminate only if / is zero or a
positive integer. Even then only one of the two series converges at x = ±1.
If / is even (odd), then only the a = (a = 1) series terminates.* The
polynomials in each case have x l as their highest power of x, the next
highest being x l ~ 2 , and so on, down to x° (x) for / even (odd). By convention
these polynomials are normalized to have the value unity at x = +1 and
are called the Legendre polynomials of order /, P t (x). The first few
Legendre polynomials are :
P (x) = 1
P^x) = X
P 2 (x) = i(3x*  1)
p 3 (x) = \(5x*  3x)
p^x) = $(35a*  30z 2 + 3)
(3.15)
By manipulation of the power series solutions (3.11) and (3.14) it is
possible to obtain a compact representation of the Legendre polynomials,
known as Rodriguez' formula:
P t (x) = 4 — 7 (^l) J (316)
2 l l\dx l
[This can be obtained by other, more elegant means, or by direct /fold
integration of the differential equation (3.10).]
The Legendre polynomials form a complete orthogonal set of functions
on the interval — 1 < x < 1. To prove the orthogonality we can appeal
directly to the differential equation (3.10). We write down the differential
equation for P z (x), multiply by P v (x), and then integrate over the interval:
P Pi4*)\t(<\  * 2 ) ~) + '(' + WO*)
J— i \dx\ ax'
dx = (3.17)
* For example, if / = the a == 1 series has a general coefficient a, = ajj + 1 for
/ = 0, 2, 4 Thus the series is a (x + $x 3 + \x* + • • • .) This is just the power
es at x = ± 1 .
/l + x\
i expansion of a function Q^ix) — % In I I , which clearly diverg
For each / value there is a similar function Qi(x) with logarithms in it as the partner to
the wellbehaved polynomial solution. See Magnus and Oberhettinger, p. 59.
58 Classical Electrodynamics
Integrating the first term by parts, we obtain
Ii [ (X * " 1} S S + /(/ + W'C*)^*)] dx = ° ( 3  18 >
If we now write down (3. 1 8) with / and /' interchanged and subtract it from
(3.18), the result is the orthogonality condition:
[/(/ + 1)  /'(/' + 1)] J' P v (x)Plx) dx = (3.19)
For / ^ /', the integral must vanish. For / = /', the integral is finite. To
determine its value it is necessary to use an explicit representation of the
Legendre polynomials, e.g., Rodrigues' formula. Then the integral is
explicitly:
/ W <** = ^ j]£^  v£v  v*
Integration by parts / times yields the result:
J> ( * )]2 dx = SS J>  *&*  v**
The differentiation of (x 2 — l) 1 2/ times yields the constant (2/)!, so that
jV*(*)] 2 dx = 2&L j\l  x y dx (3.20)
The remaining integral is easily shown to be 2 2Z+1 (/!) 2 /(2/ + 1)! Con
sequently the orthogonality condition can be written :
i
P v {x)P t {x) dx = ^—^ d vi
i 21 + 1
and the orthonormal functions in the sense of Section 2.9 are
Ufc)= Jl^Pfc)
(3.21)
(3.22)
Since the Legendre polynomials form a complete set of orthogonal
functions, any function /(a;) on the interval — 1 < x < 1 can be expanded in
+1
li
! l
l
Fig. 3.2
[Sect. 3.2] BoundaryValue Problems in Electrostatics: II
terms of them. The Legendre series representation is:
oo
f(x) = ^A l P l (x)
1 =
where
A r =
21 +
! L
f(x)P,(x) dx
59
(3.23)
(3.24)
As an example, consider the function shown in Fig. 3.2:
f(x) = +1 for#>0
= 1 forz<0
Then 2l +
/ + 1 r r 1 r°
1Z_ Pi{x )dx\ P t (x)dx
2 LJo Ji J
Since P t (x) is odd (even) about x = if / is odd (even), only the odd /
coefficients are different from zero. Thus, for / odd,
A l = (21 + 1) ( 1 P 1 (x) dx (3.25)
Jo
By means of Rodrigues' formula the integral can be evaluated, yielding
,, = (r" ,2 (2 '^^ 2)!! 0.26)
where (2n + 1)!! = (2n + l)(2n  \)(2n 3)x5x3xl. Thus the
series for /(#) is:
/(*) = fP^)  P 3 (*) + HP 5 (*) ~ • ' ' (3.27)
Certain recurrence relations among Legendre polynomials of different
order are useful in evaluating integrals, generating higherorder poly
nomials from lowerorder ones, etc. From Rodrigues' formula it is a
straightforward matter to show that
dP l+1 dP,_!
 (21 + l)P t =
(3.28)
dx dx
This result, combined with differential equation (3.10), can be made to
yield various recurrence formulas, some of which are :
(/ + 1)P I+1  (21 + l)xP t + /P,! =
d Il±2 x— l (l + 1)P ? =
dx dx
dP
dx
(3.29)
60
Classical Electrodynamics
As an illustration of the use of these recurrence formulas consider the
evaluation of the integral:
h=j_ *Pt
(x)P t ix) dx
(3.30)
From the first of the recurrence formulas (3.29) we obtain an expression
for xP(x). Therefore (3.30) becomes
h = ~ { f^PfcW + 1)JW*) + WW*)] dx
The orthogonality integral (3.21) can now be employed to show that the
integral vanishes unless V = I ± 1, and that, for those values,
/:
2(1 + 1)
xP z (x)P v (x) dx = «
(21 + 1)(2/ + 3)
2/
(21  1X2/ + 1) '
I' = 1 + 1
V = 11
(3.31)
These are really the same result with the roles of / and /' interchanged. In
a similar manner it is easy to show that
x 2 Plx)P v (x) dx =
2(1 + !)(/ + 2)
(21 + 1X2/ + 3X2/ + 5)
2(2/ 2 + 2/  1)
(2/  1X2/ + 1X2/ + 3)
where it is assumed that /' > /.
r
, V = 1 + 2
" = I
(3.32)
3.3 BoundaryValue Problems with Azimuthal Symmetry
From the form of the solution of Laplace's equation in spherical
coordinates (3.2) it will be seen that, for a problem possessing azimuthal
symmetry, m = in (3.5). This means that the general solution for such
a problem is :
®{r, Q)=2 [A t r l + BfWjP^cos 6)
1 =
(3.33)
The coefficients A l and B x can be determined from the boundary condi
tions. Suppose that the potential is specified to be V(d) on the surface of a
sphere of radius a, and it is required to find the potential inside the sphere.
If there are no charges at the origin, the potential must be finite there.
Consequently B t = for all /. The coefficients A h are found by evaluating
[Sect. 3.3] BoundaryValue Problems in Electrostatics: II 61
(3.33) on the surface of the sphere :
V(d) =  Aja l Pfco& 0) (3.34)
1 =
This is just a Legendre series of the form (3.23), so that the coefficients At
are:
21 +
A,=
2a l Jo
 I V(0)P,(cos 0) sin dd (3.35)
Jo
If, for example, V(6) is that of Section 2.8, with two hemispheres at equal
and opposite potentials,
V(G) =
+ V, O<0<
2 (3.36)
V, <0<tt
2
then the coefficients are proportional to those in (3.27). Thus the potential
inside the sphere is :
0(r, 6)=V
r a PM l(3 PM + U$ p * icose) 
(3.37)
To find the potential outside the sphere we merely replace (r/a) 1 by {ajr) l+1 .
The resulting potential can be seen to be the same as (2.33), obtained by
another means.
Series (3.33), with its coefficients determined by the boundary conditions,
is a unique expansion of the potential. This uniqueness provides a means
of obtaining the solution of potential problems from a knowledge of the
potential in a limited domain, namely on the symmetry axis. On the
symmetry axis (3.33) becomes (with z = r):
0>(z = r) = f [A z r l + B*r (l+1) ] (3.38)
1 =
valid for positive z. For negative z each term must be multiplied by (— 1)'.
Suppose that, by some means, we can evaluate the potential 0(z) at an
arbitrary point z on the symmetry axis. If this potential function can be
expanded in a power series in z = r of the form (3.38), with known
coefficients, then the solution for the potential at any point in space is
obtained by multiplying each power of r l and r~ (I+1> by ^(cos 0).
62
Classical Electrodynamics
At the risk of boring the reader we return to the problem of the hemi
spheres at equal and opposite potentials. We have already obtained the
series solution in two different ways, (2.33) and (3.37). The method just
stated gives a third way. For a point on the axis we have found the closed
form (2.28):
<D(z = r )=V
1 
r" — a
rjr 2 + a 2 l
This can be expanded in powers of a 2 /r c '
* (z = r) = JL J ( _ iri «/  IW  1) («
V j = i
J
fio
(3.39)
Comparison with expansion (3.38) shows that only odd / values
(/ = 2/ — 1) enter. The solution, valid for all points outside the sphere,
is consequently:
<D(r, 6) = £ f ( 1)'"* (2j ~ VTV ~ *> (2f P^Ccos 0) (3.40)
This is the same solution as already obtained, (2.33) and (3.37).
An important expansion is that of the potential at x due to a unit point
charge at x' :
X — X
T n r>
(3.41)
1 =
where r< (/>) is the smaller (larger) of x and x', and y is the angle
between x and x', as shown in Fig. 3.3. This can be proved by rotating
axes so that x' lies along the z axis. Then the potential satisfies Laplace's
equation, possesses azimuthal symmetry, and can be expanded according
to (3.33), except at the point x = x':
00
— = 2 < A ** + B i r ~ il+1) ) p i(cos y) (3.42)
x — x
1 =
[Sect. 3.3]
BoundaryValue Problems in Electrostatics: II
63
If the point x is on the z axis, the righthand side reduces to (3.38), while
the lefthand side becomes :
1
1
1
x — x
(r 2 + r' 2  2rr' cos y) A \r  r'\
Expanding (3.43), we find
 x'l r^ 4i \rJ
(3.43)
(3.44)
For points off the axis it is only necessary, according to (3.33) and (3.38),
to multiply each term in (3.44) by i>j(cos y). This proves the general result
(3.41).
Another example is the potential due to a total charge q uniformly
distributed around a circular ring of radius a, located as shown in Fig. 3.4,
with its axis the z axis and its center at z = b. The potential at a point P
on the axis of symmetry with z = r is just q divided by the distance AP:
(J>(z = r) ==
(r 2 + c 2 — 2cr cos a)
M
(3.45)
where c 2 = a 2 + b 2 and a = tan 1 (a/b). The inverse distance AP can be
expanded using (3.41). Thus, for r > c,
00 l
®(z=r)=q^ J l ri P l (cos<x)
(3.46)
Fig. 3.4 Ring of charge of radius a and total
charge q located on the z axis with center at
z = b.
64 Classical Electrodynamics
For r < c, the corresponding form is :
<D(z = r) = <?2 TTx ^( cos <*) ( 3 47)
The potential at any /ra/nf in space is now obtained by multiplying each
member of these series by P{(cos 6) :
00 ,
<D(r, 0) = q ]? ^1 P i( cos a ) P *( cos ) ( 3  48 )
—. r i+
i=o r >
where r < (r > ) is the smaller (larger) of r and c.
3.4 Associated Legendre Polynomials and the Spherical Harmonics
So far we have dealt with potential problems possessing azimuthal
symmetry with solutions of the form (3.33). These involve only ordinary
Legendre polynomials. The general potential problem can, however, have
azimuthal variations so that m ^ in (3.5) and (3.9). Then we need the
generalization of ^(cos 6), namely, the solution of (3.9) with / and m both
arbitrary. In essentially the same manner as for the ordinary Legendre
functions it can be shown that in order to have finite solutions on the
interval — 1 < x < 1 the parameter / must be zero or a positive integer and
that the integer m can take on only the values — /, — (/ — 1), . . . , 0, . . . ,
(/ — 1), /. The solution having these properties is called an associated
Legendre function P^ix). For positive m it is defined by the formula*:
d m
Pi m (x) = ( l) w (l  * 2 r /2 ~^ Pfc) (349)
If Rodrigues' formula is used to represent P t (x), a definition valid for both
positive and negative m is obtained :
p^(x) = { —^ (1  x 2 ) m/2 ^—— (x 2  l) 1 (3.50)
i v j 2l/ , v dx l+m .
* The choice of phase for Pi m (x) is that of Magnus and Oberhettinger, and of E. U.
Condon and G. H. Shortley in Theory of Atomic Spectra, Cambridge University Press
(1953). For explicit expressions and recursion formulas, see Magnus and Oberhettinger,
p. 54.
[Sect. 3.4] BoundaryValue Problems in Electrostatics: II 65
Pr m (x) and P, m (aO are proportional, since differential equation (3.9)
depends only on m 2 and m is an integer. It can be shown that
Pf "(*) = (~D m TTT^T, JVC*) ( 3  51 >
(/ + m)\
For fixed m the functions P™(x) form an orthogonal set in the index /
on the interval — 1 < x < 1 . By the same means as for the Legendre
functions the orthogonality relation can be obtained:
i.
1 2 (I 4 mV
P,,™(x)P, m (x) dx = —^— yi ^ mh b vi (3.52)
' 2/ + l(/m)!
The solution of Laplace's equation was decomposed into a product of
factors for the three variables r, 6, and <f>. It is convenient to combine the
angular factors and construct orthonormal functions over the unit sphere.
We will call these functions spherical harmonics, although this terminology
is often reserved for solutions of the generalized Legendre equation (3.9).
Our spherical harmonics are sometimes called "tesseral harmonics" in
older books. The functions Q m (4d = eim * form a complete set of ortho
gonal functions in the index m on the interval < <f> <: 2tt. The functions
Pj m (cos 6) form a similar set in the index / for each m value on the interval
— 1 < cos 6 < 1. Therefore their product P™Q m will form a complete
orthogonal set on the surface of the unit sphere in the two indices /, m.
From the normalization condition (3.52) it is clear that the suitably
normalized functions, denoted by Y lm {6, <f>), are :
Y lm (e, & = Jl 1 ^ ( /~ m) '; paccs ey+ (3.53)
v 477 (/ + m)!
From (3.51) it can be seen that
Vi»^) = (1)X(M) (354)
The normalization and orthogonality conditions are
Jo Jo
sin 6 d6 Y* m ,(d, <f>)Y lm (0, <f>) = d r fi m . m (3.55)
The completeness relation, equivalent to (2.41), is
00 I
I I Y*Jd',<t>')Y lm {e,<i>) = d{<f>4>')d(co^Bcose') (3.56)
1=0 m= —I
For a few small / values and m > the table shows the explicit form of
the Y lm (6, <f>). For negative m values (3.54) can be used.
66
/ =
/ = 1
Classical Electrodynamics
Spherical harmonics Y lm (d, <j>)
Y J
V4tt
Jl sin 0g<*
Y„ =
r in = /_ cos o
/ = 2
y_ 2 = 111 sin 2 0<? 2i *
4V2r
K, =  /!£ sin0 cos 0e'*
r 2 „= /l/rCOS 2 0i
/ = 3
4t7\2
r 33 =   /M sin 3 0e**
4 V 4tt
1 /105 .
sin* cos
y =  i /il sin 0(5 cos 2  l)e<*
4 V 4tt
Yon = fl.( cos 3 6   cos
4tt \2
I  cos 3 —  COS )
\2 2 J
Note that, for m = 0,
Y l0 (d,<f>) = J^^P l ( C os&)
(3.57)
An arbitrary function g(6, </>) can be expanded in spherical harmonics :
(3.58)
00 I
g(o, <t>) = I 2 i4 lm r Im (0, 0)
1 = m — —I
where the coefficients are
A lm =jdQ Y t *(e, 4>)g{Q, <f>)
[Sect. 3.5] BoundaryValue Problems in Electrostatics: II
67
A point of interest to us in the next section is the form of the expansion
for = 0. With definition (3.57), we find:
oo
21 + 1
where
m
dQP l (cosd)g(6, ( f>)
(3.59)
(3.60)
All terms in the series with m ^ vanish at 6 = 0.
The general solution for a boundaryvalue problem in spherical coordi
nates can be written in terms of spherical harmonics and powers of r in a
generalization of (3.33):
<D(r,0,<£)=2 J [A lm r l + B lm r« +1 ^Y lm (d, <f>) (3.61)
1 = m ■= —I
If the potential is specified on a spherical surface, the coefficients can be
determined by evaluating (3.61) on the surface and using (3.58).
3.5 Addition Theorem for Spherical Harmonics
A mathematical result of considerable interest and use is called the
addition theorem for spherical harmonics. Two coordinate vectors x and
x', with spherical coordinates (r, 6, <f>) and (r', 6', <f>'), respectively, have an
angle y between them, as shown in Fig. 3.5. The addition theorem
expresses a Legendre polynomial of order / in the angle y in terms of
Fig. 3.5
68 Classical Electrodynamics
products of the spherical harmonics of the angles 6, cf> and 6', cf>' :
+
i
Piioos y) = ^j^ J Y <*( '' WimiO, <f>) (3.62)
where cos y = cos 6 cos 0' + sin 6 sin 0' cos (<£ — <f>'). To prove this
theorem we consider the vector x' as fixed in space. Then Pj(cos y) is a
function of the angles 0, cf>, with the angles 6', $' as parameters. It may be
expanded in a series (3.58):
P,(cosy) = i J A v JB',<f>')Y v Jd,<i>) (3.63)
l'=0 m= —V
Comparison with (3.62) shows that only terms with /' = / appear. To see
why this is so, note that, if coordinate axes are chosen so that x' is on the z
axis, then y becomes the usual polar angle and Pj(cos y) satisfies the
equation:
V' 2 P,(cos y) + 1 1L±J± Pi(cos y) = o (3.64)
r*
where V' 2 is the Laplacian referred to these new axes. If the axes are now
rotated to the position shown in Fig. 3.5, V' 2 = V 2 and r is unchanged.*
Consequently Pj(cos y) still satisfies an equation of the form (3.64); i.e.,
it is a spherical harmonic of order /. This means that it is a linear com
bination of 7j TO 's of that order only:
P l (cosy)= i A m id',<j>')Y lm {B,<f>) (365)
m= —I
The coefficients A m (6', <f>') are given by:
ajo', </>') = \y* m {e, cf>)p l ( C os y) da (3.66)
To evaluate this coefficient we note that it may be viewed, according to
(3.60), as the m' = coefficient in an expansion of the function
V4tt/(21 + 1) Yfijd, <f>) in a series of Y lmf {y, ft) referred to the primed
axis of (3.64). From (3.59) it is then found that, since only one / value is
present, coefficient (3.66) is
AJQ', V) = ^^ [Y* m (0(y, ft, <i>{y, 0))], =o (3.67)
In the limit y — > 0, the angles (0, <f>), as functions of (y, /?), go over into
* The proof that V' 2 = V 2 under rotations follows most easily from noting that
V 2 y» = V • Vy) is an operator scalar product, and that all scalar products are invariant
under rotations.
[Sect. 3.6] BoundaryValue Problems in Electrostatics: II 69
(6', <f>'). Thus addition theorem (3.62) is proved. Sometimes the theorem
is written in terms of P t m (co& 6) rather than Y lm . Then it has the form:
Pj(cos y) = P ? (cos 0)Pj(cos 0')
i
+ 2 S (/ ~ m)! P ™(cos 6)P l m (cos 0') cos [m(<£  <f>' )] (3.68)
It the angle y goes to zero, there results a "sum rule" for the squares of
Z irimC^r^r 11 < 3  69)
*' , Air
m= — I
The addition theorem can be used to put expansion (3.41) of the potential
at x due to a unit charge at x' into its most general form. Substituting
(3.62) for Pj(cos y) into (3.41) we obtain
ir^Ti = ""2 2 jTTT 5^i Y '* (e: ^^ « (3  70)
l X ~ X I 1=0 m=l Zl + lr >
Equation (3.70) gives the potential in a completely factorized form in the
coordinates x and x'. This is useful in any integrations over charge
densities, etc., where one variable is the variable of integration and the
other is the coordinate of the observation point. The price paid is that
there is a double sum involved, rather than a single term.
3.6 Laplace's Equation in Cylindrical Coordinates; Bessel Functions
In cylindrical coordinates (p, <f>, z), as shown in Fig. 3.6, Laplace's
equation takes the form :
d P 2 pdp p 2 dcf> 2 dz 2 v '
The separation of variables is accomplished by the substitution :
0( P , <M = P0>)fi(#Z( Z ) (372)
In the usual way this leads to the three ordinary differential equations :
dp 2 p dp
d 2 7
—  k 2 Z =
dz 2
(3.73)
<^ + v 2 Q =
dj> 2 *
(3.74)
fc 2 ]W = o
(3.75)
70
Classical Electrodynamics
Fig. 3.6
The solutions of the first two equations are elementary :
Z(z) = e ±kz
(2(0) = e ±iv *
(3.76)
In order that the potential be single valued, v must be an integer. But
barring some boundarycondition requirement in the z direction, the
parameter k is arbitrary. For the present we will assume that k is real.
The radial equation can be put in a standard form by the change of
variable x = kp. Then it becomes
'* + ii« + (i_r, R _o
dx 2
dx
(3.77)
This is Bessel's equation, and the solutions are called Bessel functions of
order v. If a power series solution of the form:
R(x) = x a ^ ajx j
3=0
is assumed, then it is found that
and
a = ±v
1
«2/ = 
4/0' + ^
«2j2
(3.78)
(3.79)
(3.80)
for j = 0, 1, 2, 3, ... . All odd powers of x j have vanishing coefficients.
The recursion formula can be iterated to obtain
«2i =
(lffla+1)
2 2 V!T0 + a+l)
(3.81)
[Sect. 3.6] BoundaryValue Problems in Electrostatics: II 71
It is conventional to choose the constant a = [2 x F(y. + l)] 1 . Then the
two solutions are
^)=(lS /^ (f 0.82)
\2/^;!ro + v + i)\2/
These solutions are called Bessel functions of the first kind of order ±v.
The series converge for all finite values of x. If v is not an integer, these
two solutions J ±v (x) form a pair of linearly independent solutions to the
secondorder Bessel's equation. However, if v is an integer, it is well known
that the solutions are linearly dependent. In fact, for v = m, an integer,
it can be seen from the series representation that
/*(*) = (D m / m (*) (3.84)
Consequently it is necessary to find another linearly independent solution
when m is an integer. It is customary, even if v is not an integer, to replace
the pair J ±v (x) by J v (%) and N v (x), the Neumann function (or Bessel's
function of the second kind) :
N v (x) = •^cosvirJ.^) (3 g5)
sin vn
For v not an integer, N v (%) is clearly linearly independent of J v (x). In the
limit v ► integer, it can be shown that N 9 (x) is still linearly independent
of J v (x). As expected, it involves log a;. Its series representation is given
in the reference books.
The Bessel functions of the third kind, called Hankel functions, are
defined as linear combinations of /„(x) and N v (x):
H?\x) = J v (x) + iN v (*)]
I (386)
H™(x) = JXx)iN v (x)j
The Hankel functions form a fundamental set of solutions to Bessel's
equation, just as do J v (x) and N v (x).
The functions J v , N v , H ( }\ H™ all satisfy the recursion formulas:
Q. v  1 {x) + n v+1 0) = — Q v O) (3.87)
x
n v ^(x)  n v+1 (x) = 2 ?%^ (3<88)
ax
72
Classical Electrodynamics
where Q V ( x ) is an y one °f the cylinder functions of order v. These may be
verified directly from the series representation (3.82).
For reference purposes, the limiting forms of the various kinds of
Bessel functions will be given for small and large values of their argument.
Only the leading terms will be given for simplicity :
x < 1 J v ( x ) —>
1
N v ( x )
.5772
I .
TT \X/
In these formulas v is assumed to be real and nonnegative.
^ . 7 / X /^ ( VTT Tt\~
cc>1, v J v (a;)> / — cos I a; 1
N ttx \ 2 4/
•
N v (x) > /— sin I x  —  I
N ttx \ 2 4/ J
(3.89)
(3.90)
(3.91)
The transition from the small x behavior to the large x asymptotic form
occurs in the region of x ~ v.
From the asymptotic forms (3.91) it is clear that each Bessel function
has an infinite number of roots. We will be chiefly concerned with the
roots of J v { x ) '•
J v (xJ = 0, n = 1, 2, 3, . . . (3.92)
x vn is the nth root of J£x). For the first few integer values of v, the first
three roots are:
v = 0, x 0n = 2.405, 5.520, 8.654
v= 1,
= 3.832,7.016, 10.173,...
v = 2, x 2n = 5.136, 8.417, 11.620, . . .
For higher roots, the asymptotic formula
x vn ^ riTT + (y  i) "
gives adequate accuracy (to at least three figures). Tables of roots are
given in Jahnke and Emde, pp. 166168.
Having found the solution of the radial part of Laplace's equation in
terms of Bessel functions, we can now ask in what sense the Bessel
functions form an orthogonal, complete set of functions. We will consider
[Sect. 3.6] BoundaryValue Problems in Electrostatics: II 73
only Bessel functions of the first kind, and will show that Vp J v (x vn p/a), for
fixed v > 0, n = 1, 2, . . . , form an orthogonal set on the interval <
p < a. The demonstration starts with the differential equation satisfied by
J (x old):
If we multiply the equation by p/ v (# vn ,p/a) and integrate from to a, we
obtain
+
Integration by parts, combined with the vanishing of (pJJ v ') at p =
(for v > 0) and p = a, leads to the result :
^0 i/p ^0
^P dp
+
ff^JW^fW^f)"'
If we now write down the same expression, with n and ri interchanged,
and subtract, we obtain the orthogonality condition :
(fin ~ x2 vn')j Q Pjy^vn' £ j^v(*v« ") dp = (3.94)
By means of the recursion formulas (3.87) and (3.88) and the differential
equation, the normalization integral can be found to be :
1 pJv \ vn ' a) Jv V vn al dp = ^ [ J v + i(*v„)] 2<5 «'n ( 3  9 5)
Assuming that the set of Bessel functions is complete, we can expand an
arbitrary function of p on the interval ^ p < a in a BesselFourier
series :
/(p) = ^A vn J v (x vn £) (3.96)
74
where
Classical Electrodynamics
A ™ = 2T2 2 , , fVoW^) dp (3.97)
Our derivation of (3.96) involved the restriction v > 0. Actually it can
be proved to hold for all v > — 1 .
Expansion (3.96) and (3.97) is the conventional FourierBessel series
and is particularly appropriate to functions which vanish at p = a (e.g.,
homogeneous Dirichlet boundary conditions on a cylinder; see the
following section). But it will be noted that an alternative expansion is
possible in a series of functions \^~pJ v (ymPl a ) where y vn is the nth root of
the equation [dJ v {x)]jdx = 0. The reason is that, in proving the ortho
gonality of the functions, all that is demanded is that the quantity
[pJ v (Xp)(d/dp)J v (X' p)] vanish at the end points p = and p = a. The
requirement is met by either % = x v Ja or X = y^/a, where J v ( x vr) = an( ^
J v '(y V n) — 0. The expansion in terms of the set V pJ v {y vn p\a) is especially
useful for functions with vanishing slope at p = a. (See Problem 3.8.)
A FourierBessel series is only one type of expansion involving Bessel
functions. Neumann series
2 a Jv
n=0
S z )
Kapteyn series
rc=0
Jv+n(( v + "»
, and Schlomilch series
2 aj v {nx)
Ln = l
are some of the other
possibilities. The reader may refer to Watson, Chapters XVIXIX, for a
detailed discussion of the properties of these series. Kapteyn series occur
in the discussion of the Kepler motion of planets and of radiation by
rapidly moving charges (see Problems 14.7 and 14.8).
Before leaving the properties of Bessel functions it should be noted that
if, in the separation of Laplace's equation, the separation constant k 2 in
(3.73) had been taken as —k 2 , then Z(z) would have been sin kz or cos kz
and the equation for R(p) would have been :
d 2 R ldR
dp 2 p dp
/c 2 +  2 K =
With kp = x, this becomes
dx 2 x dx \ x 2 /
(3.98)
(3.99)
The solutions of this equation are called modified Bessel functions. It is
evident that they are just Bessel functions of pure imaginary argument.
[Sect. 3.7] BoundaryValue Problems in Electrostatics: II
75
The usual choices of linearly independent solutions are denoted by I v (x)
and K v (x). They are defined by
I£x) = r v J v (ix)
K v (x) =  i^H^Xix)
(3.100)
(3.101)
and are real functions for real x. Their limiting forms for small and large
x are, assuming real v > :
x<l J v (;r)^1 h)
~(v + 1)\2/
*v(*)
(lng) + 0.5772..),
L 2 \z/'
1
x > 1, v I v (x) > —L= e a
yJllTX
K v (x)
— e '
2x
1 +
1 +
v =
j^0
(3.102)
(3.103)
(3.104)
3.7 BoundaryValue Problems in Cylindrical Coordinates
The solution of Laplace's equation in cylindrical coordinates is $ =
R(p)Q(<f>)Z(z), where the separate factors are given in the previous section.
Consider now the specific boundary value problem shown in Fig. 3.7.
The cylinder has a radius a and a height L, the top and bottom surfaces
being at z = L and z = 0. The potential on the side and the bottom of
the cylinder is zero, while the top has a potential O = V(p, <f>). We want
to find the potential at any point inside the cylinder. In order that O be
single valued and vanish at z ■■= 0,
Q(<f>) = A sin m<f> + B cos m<j>
Z(z) = sinh kz
(3.105)
where v = m is an integer and k is a constant to be determined. The radial
factor is
R(p) = CJJkp) + DN m (kp)
(3.106)
16
Classical Electrodynamics
X
# = o
■f =V(p,<t>)
* =
Fig. 3.7
If the potential is finite at p = 0, D = 0. The requirement that the
potential vanish at /> = a means that k can take on only those special
values :
fc WM = ^, n = 1,2,3,... (3.107)
where jc mw are the roots ofJ m (x mn ) = 0.
Combining all these conditions, we find that the general form of the
solution is
® (p, <M = 2 1 JmiKnP) sinh {k mn z)[A mn sin m<f> + B mn cos m<f>]
m=0 n = l
(3.108)
At z = L, we are given the potential as V(p, <f>). Therefore we have
V(p> <f>) = 2 sinn (.k mn L)J m (k mnP )[A mn sin m<f> + B mn cos w0]
m,n
This is a Fourier series in <f> and a BesselFourier series in p. The coeffi
cients are, from (2.43) and (3.97),
_ 2 cosech
mn ,2 r2
and
TTCfJ
2 cosech
Bwm Tra 2 ^
;h(kwwL ^ f *<ty f °dp pF(p, WJkmnP) sin m<£
+ l( fc «n fl ) ■'O *>0
:ll(fcmwL) P^ f'dp PHP, WmikmnP) COS m<f>
+i(k mn a) Jo Jo
(3.109)
with the proviso that, for m = 0, we use fB^ in the series.
The particular form of expansion (3.108) is indicated by the requirement
that the potential vanish at z = for arbitrary p and at p = a for arbitrary
z. For different boundary conditions the expansion would take a different
[Sect. 3.8] BoundaryValue Problems in Electrostatics: II 11
form. An example where the potential is zero on the end faces and equal
to V(<f>, z) on the side surface is left as Problem 3.6 for the reader.
The FourierBessel series (3.108) is appropriate for a finite interval in
P, < p < a. If a > oo, the series goes over into an integral in a manner
entirely analogous to the transition from a trigonometric Fourier series
to a Fourier integral. Thus, for example, if the potential in chargefree
space is finite for z > and vanishes for z — ► oo, the general form of the
solution for z > must be
oo /»
<D0, <f>, z) = y \dk e kz JJk P )[AJk) sin m<f> + BJk) cos m<£] (3.110)
If the potential is specified over the whole plane z = to be V(p, <f>) the
coefficients are determined by
oo <»
v (p, <f>) = y\ dkJJkp)[A m (k) sin m<f> + BJk) cos m<f\
The variation in (f> is just a Fourier series. Consequently the coefficients
A m (k) and B m (k) are separately specified by the integral relations :
WJo Icosm^J Jo U m (fe')J
These radial integral equations of the first kind can be easily solved, since
they are Hankel transforms. For our purposes, the integral relation,
xJJkx)JJk'x) dx = \ d(k'  k) (3.112)
Jo k
can be exploited to invert equations (3.111). Multiplying both sides by
P J J^p) and integrating over p, we find with the help of (3.112) that the
coefficients are determined by integrals over the whole area of the plane
z = 0:
A m( k )} kC 00 C 2 * fsinm<A ^^^
R J = £ d P p \ WfaWJlW \ (3 ' 113)
BmW) nJ o Jo [cos m(f>
As usual, for m = 0, we must use $B (k) in series (3.110).
3.8 Expansion of Green's Functions in Spherical Coordinates
In order to handle problems involving distributions of charge as well as
boundary values for the potential (i.e., solutions of Poisson's equation) it
is necessary to determine the Green's function G(x, x') which satisfies the
78
Classical Electrodynamics
appropriate boundary conditions. Often these boundary conditions are
specified on surfaces of some separable coordinate system, e.g., spherical or
cylindrical boundaries. Then it is convenient to express the Green's
function as a series of products of the functions appropriate to the coordi
nates in question. We first illustrate the type of expansion involved by
considering spherical coordinates.
For the case of no boundary surfaces, except at infinity, we already
have the expansion of the Green's function, namely (3.70) :
x — x
00 _ I
=4 "2 2
1
1 = m = I
21 + 1 r\
Y?JP'> Wm(0»
Suppose that we wish to obtain a similar expansion for the Green's
function appropriate for the "exterior" problem with a spherical boundary
at r = a. The result is readily found from the image form of the Green's
function (2.22). Using expansion (3.70) for both terms in (2.22), we obtain :
G(x, x') = 4tt y — !—
K ' f" 2/ + 1
l,m '
r 1 ? 1 a\rr J J
To see clearly the structure of (3.114) and to verify that it satisfies the
boundary conditions, we exhibit the radial factors separately for r < r'
and for r > /:
„2l + l~
1
_rl +1 a\rr'J
2\* + l
r —
j+i
r <r'
(3.115)
3Ti» r>r '
First of all, we note that for either r or r' equal to a the radial factor
vanishes, as required. Similarly, as rorr'^ oo, the radial factor vanishes.
It is symmetric in r and r' . Viewed as a function of r, for fixed r\ the
radial factor is just a linear combination of the solutions r l and r~ {l+1) of the
radial part (3.7) of Laplace's equation. It is admittedly a different linear
combination for r < r' and for r > r' . The reason for this will become
apparent below, and is connected with the fact that the Green's function
is a solution of Poisson's equation with a delta function inhomogeneity.
Now that we have seen the general structure of the expansion of a
Green's function in separable coordinates we turn to the systematic con
struction of such expansions from first principles. A Green's function for
a potential problem satisfies the equation
W x 2 G(x, x') = 4tt <5(x  x')
(3.116)
[Sect. 3.8] BoundaryValue Problems in Electrostatics: H 79
subject to the boundary conditions G(x, x') = for either x or x' on the
boundary surface S. For spherical boundary surfaces we desire an expan
sion of the general form (3.1 14). Accordingly we exploit the fact that the
delta function can be written *
d(x  x') = \ d(r  r') d(<f>  <f>') <3(cos 6  cos 0')
r 2
and that the completeness relation (3.56) can be used to represent the
angular delta functions:
.. oo I
d(xx') = \d(rr')^ J YtnW'.WtJQ,*) ( 3  117 )
1 = m= I
Then the Green's function, considered as a function of x, can be expanded
as
oo I
G(x, x') = I I A lm (6', <f>') gl (r, r')Y lm (e, <f>) (3.H8)
1 = m= —l
Substitution of (3.117) and (3.118) into (3.116) leads to the results
A lm {e',<f>')=Y l l{d'^') (3.119)
and
Id* ( /Yk /(/ + 1) , ,. 4tt
 — 2 (rgi(r, r'))  ^— J gl (r, r ) =  
r dr 2 r* r
" 4" 2 (rgfr, r'))  ^±^ gl (r, V) =  ^ d(r  r') (3.120)
The radial Green's function is seen to satisfy the homogeneous radial
equation (3.7) for r ^ r ' . Thus it can be written as :
(Ar l + Br~ il+1 \ for r < r'
gi(r, r ) = ^ + B > r «+i\ for r > r'
The coefficients A, B, A', B' are functions of r' to be determined by the
boundary conditions, the requirement implied by d(r — r') in (3.120), and
the symmetry of gl (r, r') in r and r'. Suppose that the boundary sufaces are
concentric spheres at r = a and r = b. The vanishing of G(x, x') for x on
* To express d(x  x') = <5(*i  *i') %2  *■') K x z — x s) in terms of the coordi
nates (li, l 2 , l 3 ), related to (x u x 2 , x 3 ) via the Jacobian J(x u £*), we note that the mean
ingful quantity is <5(x — x') d 3 x. Hence
d(x  x') = n^r7 <K£i  Si') <Kf .  f .0 <5(S 3  f .0
80
Classical Electrodynamics
the surface implies the vanishing of g x {r, r') for r = a and r = b. Con
sequently g t (r, r') becomes
gi(r, r') =
a 2l+1 \
r i + i)>
B ,(J rM
\ r l + l b 2l + lJ>
r <r'
r>r'
(3.121)
The symmetry in r and r' requires that the coefficients A(r') and B'(r') be
such that g t (r, r') can be written
where r< (>•.>) is the smaller (larger) of r and r'. To determine the constant
C we must consider the effect of the delta function in (3. 1 20). If we multiply
both sides by r and integrate over the interval from r = r' — e to r = r' +
e, where e is very small, we obtain
■f(r gl (r,r'))
Ldr
^{rgi(r,r>))
r' + e Ldr
Ait
=   (3.123)
Thus there is a discontinuity in slope at r = r', as indicated in Fig. 3.8.
For r = r' + e, r > = r, r < = r' . Hence
J {rgir, r'))
Ldr
Similarly
T(^(r,r'))
Ldr
_r{, a n+x \\dl\ r' +1 \"
\ r' l+l /ldr\r l b 2l+1 I l=r>
^fen('«<n
^♦■♦■fenen
Z + 1 + /II 111
r'e r ^
Substituting these derivatives into (3.123), we find:
4tt
C =
(2/ + 1)
Hfl
(3.124)
Combination of (3.124), (3.122), (3.119), and (3.118) yields the expansion
of the Green's function for a spherical shell bounded by r = a and r = b:
z=o m=f (2/ + 1)
Mi)'
[Sect. 3.9] BoundaryValue Problems in Electrostatics: II
81
Fig. 3.8 Discontinuity in slope of
the radial Green's function.
\ 1
\ '
x 1 '
k
\
\ 1 y
Jlv^
V.
<^l\v
C
1
^ ' \^*^^
1 X
For the special cases a + 0, b > oo, and b > oo, we recover the previous
expansions (3.70) and (3.114), respectively. For the "interior" problem
with a sphere of radius b we merely let a ► 0. Whereas the expansion for
a single sphere is most easily obtained from the image solution, the general
result (3.125) for a spherical shell is rather difficult to obtain by the method
of images, since it involves an infinite set of images.
3.9 Solution of Potential Problems with the Spherical Green's Function
Expansion
The general solution to Poisson's equation with specified values of the
potential on the boundary surface is (see Section 1.10):
O(x) = I p(x')G(x, x') dV   ct O(x') ^ da' (3.126)
J v Att Js on'
For purposes of illustration let us consider the potential inside a sphere of
radius b. First we will establish the equivalence of the surface integral in
(3.126) to the previous method of Section 3.4, equations (3.61) and (3.58).
With a = in (3.125), the normal derivative, evaluated at r' = b, is:
dG
dn'
dG
dr'
,_ 6 = ~ j^{i) 1yU6 '' <m ™ (0 ' ^ (3  127)
l,m
Consequently the solution of Laplace's equation inside r = b with
$ = V{0', ^') on the surface is, according to (3.126):
<D(x)
l,m
)YtJd',<f>')dO.'
(;)'
Y lm (d, <f>) (3.128)
For the case considered, this is the same form of solution as (3.61) with
(3.58). There is a third form of solution for the sphere, the socalled
82
Classical Electrodynamics
Fig. 3.9 Ring of charge of radius a and
total charge Q inside a grounded, conduct
ing sphere of radius b.
Poisson integral (2.25). The equivalence of this solution to the Green's
function expansion solution is implied by the fact that both were derived
from the general expression (3.126) and the image Green's function. The
explicit demonstration of the equivalence of (2.25) and the series solution
(3.61) will be left to the problems.
We now turn to the solution of problems with charge distributed in the
volume, so that the volume integral in (3.126) is involved. It is sufficient
to consider problems in which the potential vanishes on the boundary
surfaces. By linear superposition of a solution of Laplace's equation the
general situation can be obtained. The first illustration is that of a hollow
grounded sphere of radius b with a concentric ring of charge of radius a
and total charge Q. The ring of charge is located in the xy plane, as shown
in Fig. 3.9. The charge density of the ring can be written with the help of
delta functions in angle and radius as
p(x') = £ d(r'  a) «5(cos 6>') (3.129)
27ur
In the volume integral over the Green's function only terms in (3.125) with
m = will survive because of azimuthal symmetry. Then, using (3.57)
and remembering that a + in (3.125), we find
O(x) = J />(x')G(x, x') d 3 x'
i=o Xr> ° '
where now r < (r>) is the smaller (larger) of r and a. Using the fact that
(— l) n (2n — 1)"
An+iCO) = and P 2n (0) = ^ — — , (3.130) can be written as:
<D(x) = q2
2 n n\
■l)"(2nl)l! 2w
2 n n\
{^■0r) P ^osB) (3.131)
[Sect. 3.9] BoundaryValue Problems in Electrostatics: II
83
Fig. 3.10 Uniform line charge of
length lb and total charge Q inside
a grounded, conducting sphere of
radius b.
In the limit b^ oo, it will be seen that (3.130) or (3.131) reduces to
expression (3.48) for a ring of charge in free space. The present result can
be obtained alternatively by using (3.48) and the images for a sphere.
A second example of charge densities, illustrated in Fig. 3.10, is that of
a hollow grounded sphere with a uniform line charge of total charge Q
located on the z axis between the north and south poles of the sphere.
Again with the help of delta functions the volumecharge density can be
written :
p(x') = Q _1_. [5( C os 6'  1) + <5(cos 0' + 1)] (3.132)
2b 2nr'"'
The two delta functions in cos correspond to the two halves of the line
charge, above and below the xy plane. The factor 2nr" 1 in the denominator
assures that the charge density has a constant linear density Q\2b. With
this density in (3.126) we obtain
*(*) = Tu 2 CW) + p i(~ W p i( cos e "> [ r <{7^ ~ iiri) dr ' (3  133)
The integral must be broken up into the intervals < r' < r and
r </ < b. Then we find
1(1 +> 1)\ \b
For / = this result is indeterminate
have, for / = only,
d
dl\ \b
(3.134)
Applying L'Hospital's rule, we
P_ to ^LJeI _ Um (_ I />»<*) = ,„ (*) (3.135)
Jo io d ... io \ dl J \rl
dl
84
Classical Electrodynamics
This can be verified by direct integration in (3.133) for / = 0. Using the
fact that Pj(l) = (1)', the potential (3.133) can be put in the form:
<D(x)
?M9
+
V (4/ + 1)
^2;(2/ + l)
1  (y P*fro& 0)} (3.136)
The presence of the logarithm for / = reminds us that the potential
diverges along the z axis. This is borne out by the series in (3.136), which
diverges for cos 6 = ±1, except at r = b exactly.
The surfacecharge density on the grounded sphere is readily obtained
from (3.136) by differentiation:
«(9) i?*
477 dr
Antfl
1 + y(4L±i)p 2Xcos0)
£i(2/ + D
(3.137)
The leading term shows that the total charge induced on the sphere is — Q,
the other terms integrating to zero over the surface of the sphere.
3.10 Expansion of Green's Functions in Cylindrical Coordinates
The expansion of the potential of a unit point charge in cylindrical
coordinates affords another useful example of Green's function expan
sions. We will present the initial steps in general enough fashion that the
procedure can be readily adapted to finding Green's functions for potential
problems with cylindrical boundary surfaces. The starting point is the
equation for the Green's function :
V, 2 G(x, X ') =  — d(p  />') d(<j>  0') d(z  z') (3.138)
P
where the delta function has been expressed in cylindrical coordinates.
The <f> and z delta functions can be written in terms of orthonormal
functions :
d(z  z') = — dk e ik{z  z,) =  \ dk cos [k(z  2')]
2.7T J — 00 77 Jo
1 °° .
d(<f> —(/>') = — V £<«(♦♦')
m= — 00
(3.139)
We expand the Green's function in similar fashion :
G(x,x') = — ^ )dke in **^coslk{zz')]g m {p,p') (3.140)
«i = — on ^
85
[Sect. 3.10] BoundaryValue Problems in Electrostatics: II
Then substitution into (3.138) leads to an equation for the radial Green's
function g m (p, p'):
1 t i? ¥)  i k * + *)* =   «*  * < 3  i4i)
p dp\ dp l \ pi p
For p # p this is just equation (3.98) for the modified Bessel functions,
IJkp) and KJkp). Suppose that yjjep) is some linear combination of
I m and K m which satisfies the correct boundary conditions for p < p, and
that y) 2 (kp) is a linearly independent combination which satisfies the
proper boundary conditions for p > p. Then the symmetry of the Green's
function in /> and p requires that
gm(p, P') = V>i(kp<)y>2(kp>)
(3.142)
The normalization of the product ^ 2 is determined by the discontinuity
in slope implied by the delta function in (3.141):
dp
dp
Att
7
(3.143)
where I . means evaluated at p = p ± e. From (3.142) it is evident that
= Kv>iV>*  VWi') = k^l>i> Vs] (3.144)
dg m
 dp
_ dgrn
+ dp
where primes mean differentiation with respect to the argument, and
W{(p x ,ip^ is the Wronskian of ^ and ip 2 . Equation (3.141) is of the
SturmLiouville type
(3.145)
and it is well known that the Wronskian of two linearly independent
solutions of such an equation is proportional to [l/p(x)]. Hence the
possibility of satisfying (3.143) for all values of p is assured. Clearly we
must demand that the normalization of the product %p x tp 2 is such that the
Wronskian has the value :
x
(3.146)
If there are no boundary surfaces, the requirement is that g m (p, p) be
finite at /> = and vanish at p > oo. Consequently tp x (kp) = AI m (kp) and
ip 2 (fcp) = KJJcp). The constant A is to be determined from the Wronskian
condition (3.146). Since the Wronskian is proportional to (\jx) for all
values of x, it does not matter where we evaluate it. Using the limiting
86 Classical Electrodynamics
forms (3.102) and (3.103) for small x [or (3.104) for large x], we find
W[I m (x), K m (z)] =  i (3.147)
x
so that A = Att. The expansion of l/x — x' therefore becomes:
J—  = l J \*dk *«***"> cos [k(zz')\IJk P< )K m (k P> ) (3.148)
This can also be written entirely in terms of real functions as :
i 4 r°°
= \ dk cos [k(z  z')~]
x — x' IT Jo
x hl (k P< )K (k P> ) + y cos [m(<£  f )]J M (fe/»<)X w (kp>)
(3.149)
A number of useful mathematical results can be obtained from this
expansion. If we let x' — >■ 0, only the m = term survives, and we obtain
the integral representation :
i 2 r°°
j=L= =  cos kz K (k P ) dk (3.150)
V f> 2 + Z 2 TT J <>
If we replace /> 2 in (3.150) by R 2 = P 2 + />' 2  2 PP cos (0  <f>'\ then we
have on the lefthand side the inverse distance x — x' _1 with z' = 0, i.e.,
just (3.149) with z = 0. Then comparison of the righthand sides of
(3.149) and (3.150) (which must hold for all values of z) leads to the
identification :
K (kJ P 2 + P ' 2  2 PP ' cos (<f>  #) )
= I (k P< )K (k P> ) + 2^008 [m(<J>  cf>')]I m (k P< )K m (k P> ) (3.151)
m = l
In this last result we can take the limit k —> and obtain an expansion for
the Green's function for (twodimensional) polar coordinates :
1
In
^p 2 + p' 2  2pp' cos (tp  <p')j
= In (— ) + y (^fcos [m(<£  f )] (3.152)
This representation can be verified by a systematic construction of the
twodimensional Green's function for Poisson's equation along the lines
leading to (3.148).
[Sect. 3.11] Boundary Value Problems in Electrostatics: II 87
3.11 Eigenfunction Expansions for Green's Functions
Another technique for obtaining expansions of Green's functions is the
use of eigenf unctions for some related problem. This approach is inti
mately connected with the methods of Sections 3.8 and 3.10.
To specify what we mean by eigenfunctions, we consider an elliptic
differential equation of the form :
VVx) + [/(x) + Afy(x) = (3.153)
If the solutions y(x) are required to satisfy certain boundary conditions
on the surface S of the volume of interest V, then (3.153) will not in general
have wellbehaved (e.g., finite and continuous) solutions, except for
certain values of A. These values of A, denoted by A n , are called eigenvalues
(or characteristic values) and the solutions tpjx) are called eigenfunctions.*
The eigenvalue differential equation is written:
VV(x) + [/(x) + A> n (x) = (3.154)
By methods similar to those used to prove the orthogonality of the
Legendre or Bessel functions it can be shown that the eigenfunctions are
orthogonal :
I* </V*(x)Vn(x) <?x = d mn (3.155)
Jv
where the eigenfunctions are assumed normalized. The spectrum of
eigenvalues A w may be a discrete set, or a continuum, or both. It will be
assumed that the totality of eigenfunctions forms a complete set.
Suppose now that we wish to find the Green's function for the equation :
V x 2 G(x, x') + [fix) + A]G(x, x') = 4tt<5(x  x') (3.156)
where A is not in general one of the eigenvalues A n of (3. 1 54). Furthermore,
suppose that the Green's function is to have the same boundary conditions
as the eigenfunctions of (3.154). Then the Green's function can be
expanded in a series of the eigenfunctions of the form :
G(x,x') = 2«n(*>»(x) (3.157)
n
Substitution into the differential equation for the Green's function leads
to the result:
2 a m (x')(A  A „> m (x) = 4tt<5(x  x') (3.158)
m
* The reader familiar with wave mechanics will recognize (3.153) as equivalent to the
Schrodinger equation for a particle in a potential.
88 Classical Electrodynamics
If we multiply both sides by y> n *(x) and integrate over the volume V, the
orthogonality condition (3.155) reduces the lefthand side to one term, and
we find : *
a n (x') = ^f^ (3.159)
Consequently the eigenfunction expansion of the Green's function is :
Gfr»o^g * , yy (3  i6o>
n n
For a continuous spectrum the sum is replaced by an integral.
Specializing the above considerations to Poisson's equation, we place
/(x) = and X = in (3.156). As a first, essentially trivial, illustration
we let (3.154) be the wave equation over all space:
(V 2 + k 2 )y k (x) = (3.161)
with the continuum of eigenvalues, k 2 , and the eigenfunctions :
1 Jkx
(2*)'
These eigenfunctions have delta function normalization :
Vk(x) = 7r3s« ( 3162 >
/
Vk'*(x)Vk(x) d 3 x = «5(k  k') (3.163)
Then, according to (3.160), the infinite space Green's function has the
expansion:  ik . (x  x <)
i — = _L \d 3 k e —— (3.164)
xx' 2tt 2 J fc 2
This is just the threedimensional Fourier integral representation of
l/xx'.
As a second example, consider the Green's function for a Dirichlet
problem inside a rectangular box defined by the six planes, x = y = z = 0,
x = a, y = b, zj*£ c. The expansion is to be made in terms of eigen
functions of the wave equation :
F 2 + kf mn )y lmn (x,y,z) = (3.165)
where the eigenfunctions which vanish on all the boundary surfaces are
and
, a / 8 . 1\itx\ . imnyX . /
VWO, y,*) = *]—r sm y— J sin \yJ sin y
lmn la 2 b 2+ cV
mrz\
c I
(3.166)
[Sect. 3.12] BoundaryValue Problems in Electrostatics: II 89
The expansion of the Green's function is therefore:
oo
I ,m,n — 1
sin
x
. (Ittx\ . (Ittx'\ . [mTTy\ . lrmTy'\ . (nirz\ . jniTz'\
m U sin [—) sm Vf) sm \—) sm VT) sm VT)
(i 2 +  2 + !L 2 )
\a 2 b 2 cV
(3.167)
To relate expansion (3.167) to the type of expansions obtained in
Sections 3.8 and 3.10, namely, (3.125) for spherical coordinates and
(3.148) for cylindrical coordinates, we write down the analogous expansion
for the rectangular box. If the x and y coordinates are treated in the
manner of (6, <f>) or {<f>, z) in those cases, while the z coordinate is singled
out for special treatment, we obtain the Green's function :
16tt V • [lirx\ • (Ittx'\ . lrmry\ . (miry'
l,m = l
~ sinh (K lm z<) sinh (K lm (c  z>))
K lm sinh (K lm c)
(3.168)
must
/ / 2 m 2 \ A
where K lm = A — + — I . If (3.167) and (3.168) are to be equal, it
be that the sum over n in (3.167) is just the Fourier series representation
on the interval (0, c) of the onedimensional Green's function in z in
(3.168): , A
,„ sm I 1
sinh (K lmZ< ) sinh (K lm (c  z>)) = 2V \ c 1 ^ Imrz
K lm sinh (K lm c) c£[ R ^ + jWj 2 \ c
(3.169)
The verification that (3.169) is the correct Fourier representation is left as
an exercise for the reader.
Further illustrations of this technique will be found in the problems at
the end of the chapter.
3.12 Mixed Boundary Conditions; Charged Conducting Disc
The potential problems discussed so far in this chapter have been of the
orthodox kind in which the boundary conditions are of one type (usually
Dirichlet) over the whole boundary surface. In the uniqueness proof for
90 Classical Electrodynamics
^■■fcj
Fig. 3.11
solutions of Laplace's or Poisson's equation (Section 1.9) it was pointed
out, however, that mixed boundary conditions, where the potential is
specified over part of the boundary and its normal derivative is specified
over the remainder, also lead to welldefined, unique, boundaryvalue
problems. There is a tendency in existing textbooks to mention the
possibility of mixed boundary conditions when making the uniqueness
proof and to ignore such problems in subsequent discussion. The reason,
as we shall see immediately, is that mixed boundary conditions are much
more difficult to handle than the normal type.
To illustrate the difficulties encountered with mixed boundary con
ditions we consider the apparently simple problem of an isolated, infinitely
thin, flat, circular, conducting disc of radius a with a total charge q placed
on it, as shown in Fig. 3.1 1. The charge distributes itself over the disc in
such a way as to make its surface an equipotential. We wish to determine
the potential everywhere in space and the charge distribution on the disc.
From the geometry of the problem we see that the potential is symmetric
about the axis of the disc and with respect to the plane containing the disc.
If cylindrical coordinates are chosen with the axis of the disc as the z axis
and the origin at the center of the disc, the potential must therefore be of
the form [from (3.110)],
(D( P , z) = ( X dkf(k)e k]zl J (k P ) (3.170)
Jo
The unknown function f{k) must be determined from the boundary
conditions at z = 0. If the potential were known everywhere over the
whole z = plane,/(&) could be found by inverting the Hankel transform,
as in going from (3.110) to (3.113). Unfortunately the boundary con
ditions at z = are not that simple. For < p < a we do know that the
potential is constant at an unknown value, O = V = q/C, where C is the
capacitance of the disc. But for a < p < oo, the potential is unknown.
{Sect. 3.12] BoundaryValue Problems in Electrostatics: II 91
From symmetry, however, we know that the normal derivative of the
potential vanishes there. Thus the boundary conditions are mixed:
0>O, 0) = V, for < P < a
^0>,0) = 0,
oz
for a < p < oo
(3.171)
The connection between the potential of the disc V and the total charge q
on it will be established by the fact that at large distances (p and/or z > a)
the potential must approach qj(p 2 + z 2 ) v K From (3.170) and an identity
of Problem 3.12c this requirement can be seen to imply
\imf(k)=q (3.172)
fc>0
When boundary conditions (3.171) are applied to the general solution
(3.170), there results a pair of integral equations of the first kind:
dkf(k)J (kp) =V, for < P
dk kf(k)J (kp) = 0, for a < p < oo
(3.173)
Such pairs of integral equations, with one of the pair holding over one
part of the range of the independent variable and the other over the other
part of the range, are known as dual integral equations. The general theory
of dual integral equations is complicated and not highly developed. But
the charged disc problem and variations of it have received considerable
attention over the years. H. Weber (1873) first solved the present problem
by using certain discontinuous integrals involving Bessel functions.
Titchmarsh, p. 334, uses Mellin transforms to effect a solution of a some
what more general pair of dual integral equations. E. T. Copson [Proc.
Edin. Math. Soc. (2), 8, 14 (1947)] reduces the disc problem to an integral
equation for the surfacecharge density of the Abel type. Tranter, p. 50
and Chapter VIII, considers slight generalizations of the pair (3.173). He
introduces a systematic technique of finding the most general form satis
fying the homogeneous member of the pair and then delimiting that form
by substitution into the other equation. The WienerHopf technique can
also be used.
For our purposes it is sufficient to observe that the dual integral
equations,
r
dyg{y)J n {yx) = x r >
dy y g(y)J n (y x ) = 0>
for < x < l
for 1 < x < oo
(3.174)
92
have the solution,
Classical Electrodynamics
*)£^^>j^(fw») (3175)
In this relation y' w (?/) is the spherical Bessel function of order n (see Section
16.1). For the set of equations (3.173) the variables are x = p/a and
y = ka, while n = 0. Thus the solution is
f(k) = Va j (ka) =  Va( S ^) (3.176)
77 7T \ ka /
Remembering the connection (3.172) which determines the potential K
in terms of the charge q, we find
2 a
This shows that the capacitance of a disc of radius a is
r 2
C =  a
This value was experimentally established with remarkable precision by
Cavendish (ca. 1780) by comparing the charges on a disc and a sphere at
the same potential.
The potential anywhere in space is found from (3.170) and (3.176) to be
*0>
Jo
00 ,, sin ka _ fr
dk  *
/ca
J (fc/>)
(3.177)
Values of the potential along the axis and in the plane of the disc can be
found readily by putting p = and z = in (3.177). The results are
O(0, z) =  tan
a
l
(!)
$(/>, 0) = 
 sin x
a
77 #
2a'
(!)■
for p > a
for < p < a
For arbitrary p and z the integral can be transformed into Weber's form
of the solution :
2a
<X>(p, z) = q sin"
V(/>  a) 2 + z 2 + V(p + a) 2 + z 2 l
(3.178)
[Sect. 3.12] BoundaryValue Problems in Electrostatics: II 93
The charge density a(p) on the surface of the disc is given by
a(p) =  — — 0, 0) = ^ dk sin ka J (kp)
2tt dz lira Jo
The integral is a wellknown discontinuous integral which vanishes for
p > a. For p < a, the charge density is
o(p) = 2 , 1 (3.179)
lira Vfl 2  p 2
The (integrable) infinity in a(p) for p ► a is a mathematical singularity
which results from the assumption of an infinitely thin disc. In practice
the charge is repelled to the outer regions of a thin disc approximately
according to (3.179), but near the edge the distribution levels off to a large,
but finite, value which depends on the detailed construction of the disc.
We have discussed the charged conducting disc in cylindrical coordinates
in order to illustrate the complications of mixed boundary conditions.
For this particular example, the mixed boundary conditions can be avoided
by separating Laplace's equation in elliptic coordinates. Then the disc
can be taken to be the limiting form of an oblate spheroidal surface. See,
for example, Smythe, pp. Ill, 156, or Jeans, p. 244.
REFERENCES AND SUGGESTED READING
The mathematical apparatus and special functions needed for the solution of potential
problems in spherical, cylindrical, spheroidal, and other coordinate systems are discussed
in
Morse and Feshbach, Chapter 10.
A more elementary treatment, with wellchosen examples and problems, can be found in
Hildebrand, Chapters 4, 5, and 8.
A somewhat oldfashioned source of the theory and practice of Legendre polynomials
and spherical harmonics, with many examples and problems, is
Byerly.
For purely mathematical properties of spherical functions one of the most useful
onevolume references is
Magnus and Oberhettinger.
For more detailed mathematical properties, see
Watson, for Bessel functions,
Bateman Manuscript Project books, for all types of special functions.
Electrostatic problems in cylindrical, spherical, and other coordinates are discussed
extensively in
Durand, Chapter XI,
Jeans, Chapter VIII,
Smythe, Chapter V,
Stratton, Chapter III.
94 Classical Electrodynamics
PROBLEMS
3.1 The surface of a hollow conducting sphere of inner radius a is divided into
an even number of equal segments by a set of planes whose common line of
intersection is the z axis and which are distributed uniformly in the angle </>.
(The segments are like the skin on wedges of an apple, or the earth's
surface between successive meridians of longitude.) The segments are kept
at fixed potentials ± V, alternately.
(a) Set up a series representation for the potential inside the sphere for
the general case of 2n segments, and carry the calculation of the coefficients
in the series far enough to determine exactly which coefficients are different
from zero. For the nonvanishing terms, exhibit the coefficients as an
integral over cos 0.
(b) For the special case of n = 1 (two hemispheres) determine explicitly
the potential up to and including all terms with 1 = 3. By a coordinate
transformation verify that this reduces to result (3.37) of Section 3.3.
3.2 Two concentric spheres have radii a, b (b > a) and are divided into two
hemispheres by the same horizontal plane. The upper hemisphere of the
inner sphere and the lower hemisphere of the outer sphere are maintained
at potential V. The other hemispheres are at zero potential.
Determine the potential in the region a < r < b as a series in Legendre
polynomials. Include terms at least up to / = 4. Check your solution
against known results in the limiting cases b > oo, and a > 0.
3.3 A spherical surface of radius R has charge uniformly distributed over its
surface with a density QJAttR 2 , except for a spherical cap at the north pole,
defined by the cone 6 = a.
(a) Show that the potential inside the spherical surface can be expressed
as
ev i
o =^ >
2 Zw2/ +
1=0
 [P J+1 (cos a)  iV^COS a)] — ^(cos 0)
where, for / = 0, P^cos a) = —1. What is the potential outside?
(b) Find the magnitude and the direction of the electric field at the origin.
(c) Discuss the limiting forms of the potential (a) and electric field (b) as
the spherical cap becomes (1) very small, and (2) so large that the area
with charge on it becomes a very small cap at the south pole.
3.4 A thin, flat, conducting, circular disc of radius R is located in the xy plane
with its center at the origin, and is maintained at a fixed potential V. With
the information that the charge density on a disc at fixed potential is
proportional to (R 2 — p 2 )~ iA , where p is the distance out from the center of
the disc,
(a) show that for r > R the potential is
1=0
(b) find the potential for r < R.
[Probs. 3] BoundaryValue Problems in Electrostatics: II 95
3.5 A hollow sphere of inner radius a has the potential specified on its surface
to be $> = V(d, <j>). Prove the equivalence of the two forms of solution for
the potential inside the sphere:
, ^ a(a 2  r 2 ) f V(0', f ) dn ,
(a) <D(x) = ^ J (r i +fl ._2arcosy)»
where cos y = cos cos 0' + sin sin 0' cos (<£ — <£')•
(6) $(x) = 2 2 ^ m w y ' w(0 ' ^
i=0 OT=I
where ^ ?m = (dCl' Y* m (d', <t>')V(6' , <j>').
3.6 A hollow right circular cylinder of radius b has its axis coincident with the
z axis and its ends at z = and z = L. The potential on the end faces is
zero, while the potential on the cylindrical surface is given as V(<f>, z).
Using the appropriate separation of variables in cylindrical coordinates,
find a series solution for the potential anywhere inside the cylinder.
3.7 For the cylinder in Problem 3.6 the cylindrical surface is made of two
equal halfcylinders, one at potential V and the other at potential  V, so
that
V(<f>, z) = «
IT , IT
Kfor  <<£ <
TT , 37T
Kfor <<£ <y
(a) Find the potential inside the cylinder.
(b) Assuming L > b, consider the potential at z = L/2 as a function of P
and <f> and compare it with twodimensional Problem 2.8.
3.8 Show that an arbitrary function fix) can be expanded on the interval
< x < a in a modified FourierBessel series
fix) = 2_. A nJv\yvnA
w = l
where y m is the nth root ^p = 0, and the coefficients A n are given by
K = ~t — ^T7~ S? (x)xj '{ y "' 3 *
3.9 An infinite, thin, plane sheet of conducting material has a circular hole of
radius a cut in it. A thin, flat disc of the same material and slightly smaller
radius lies in the plane, filling the hole, but separated from the sheet by a
very narrow insulating ring. The disc is maintained at a fixed potential V,
while the infinite sheet is kept at zero potential.
(a) Using appropriate cylindrical coordinates, find an integral expression
involving Bessel functions for the potential at any point above the plane.
96 Classical Electrodynamics
(b) Show that the potential a perpendicular distance z above the center
of the disc is
® n (z) =
o(z) = v(l  * \
\ Va 2 + z 2 /
(c) Show that the potential a perpendicular distance z above the edge of
the disc is
*«(z) = \
1 ~^K{k)
tt a
where k = 2a/(z 2 + 4a 2 )^, and K(k) is the complete elliptic integral of the
first kind.
3.10 Solve for the potential in Problem 3.2, using the appropriate Green's
function obtained in the text, and verify that the answer obtained in this
way agrees with the direct solution from the differential equation.
3.11 A line charge of length 2d with a total charge Q has a linear charge density
varying as (d 2 — z 2 ), where z is the distance from the midpoint. A grounded,
conducting, spherical shell of inner radius b > d is centered at the midpoint
of the line charge.
(a) Find the potential everywhere inside the spherical shell as an
expansion in Legendre polynomials.
(b) Calculate the surfacecharge density induced on the shell.
(c) Discuss your answers to (a) and {b) in the limit that d<^b.
3.12 (a) Verify that
1 f 00
 <X/> />')= kJ m {kp)JJkp) dk
P Jo
(b) Obtain the following expansion :
1 ^ f 00
£TT^j Z dke im ^'l'V m (kp)J m (kp')e^>^
(c) By appropriate limiting procedures prove the following expansions :
oo
J (kV p +p' 2 2 PP 'cos<f>) = ^ e im +JJJcp)J m (kp')
m= — <x>
CO
e ik P cos 4 = V in e in*JJJc p )
m= co
{d) From the last result obtain an integral representation of the Bessel
function :
l C 2n
J m (x) = \ e ix cos iim^ty
2iri m J
Compare the standard integral representations.
[Probs. 3] BoundaryValue Problems in Electrostatics: II 97
3.13 A unit point charge is located at the point (?', 4>\ z') inside a grounded
cylindrical box denned by the surfaces z = 0, z = L, P = a. Show that the
potential inside the box can be expressed in the following alternative forms:
oo oo ^**V m (^V«(— )
, ,. 4 V V \ a J \ a I
x sinhp22z<lsinh
122(1, z>)
a
m=°° m=l l m \——\
® (x ' x ' )= z^
>< 2 22
m= — a> fc = l »=1
«(* *'. sin fc) sin ^)/„(f^)/„^)
M + 0'
An +l\ x mn)
Discuss the relation of the last expansion (with its extra summation) to the
other two.
3.14 The walls of the conducting cylindrical box of Problem 3.13 are all at zero
potential, except for a disc in the upper end, denned by /> = b, at potential V.
(a) Using the various forms of the Green's function obtained in Problem
3.13, find three expansions for the potential inside the cylinder.
(b) For each series, calculate numerically the ratio of the potential at
P = 0, z = L/2 to the potential of the disc, assuming b = L/4 = a/2. Try
to obtain at least twosignificantfigure accuracy. Is one series less rapidly
convergent than the others? Why?
(Jahnke and Emde have tables of J and J x on pp. 156163, I and I x on
pp. 226229, (2ln)K and (2fr)K x on pp. 236243. Watson also has
numerous tables.)
4
Multipoles, Electrostatics of
Macroscopic Media,
Dielectrics
This chapter is first concerned with the potential due to localized
charge distributions and its expansion in multipoles. The development is
made in terms of spherical harmonics, but contact is established with the
rectangular components for the first few multipoles. The energy of a
multipole in an external field is then discussed. The macroscopic equations
of electrostatics are derived by taking into account the response of atoms
to an applied field and by suitable averaging procedures. Dielectrics and
the appropriate boundary conditions are then described, and some
typical boundaryvalue problems with dielectrics are solved. Simple
classical models are used to illustrate the main features of atomic polariza
bility and susceptibility. Finally the question of electrostatic energy in the
presence of dielectrics is discussed.
4.1 Multipole Expansion
A localized distribution of charge is described by the charge density
p(x'), which is nonvanishing only inside a sphere of radius R* around some
origin. The potential outside the sphere can be written as an expansion in
spherical harmonics :
, , V V 4tt Y lm (6, <f>)
i=o m =i ^ r * r
* The sphere of radius R is an arbitrary conceptual device employed merely to divide
space into regions with and without charge.
98
[Sect. 4.1] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 99
where the particular choice of constant coefficients is made for later
convenience. Equation (4.1) is called a multipole expansion; the / =
term is called the monopole term, / = 1 is the dipole term, etc. The
reason for these names becomes clear below. The problem to be solved
is the determination of the constants q lm in terms of the properties of the
charge density p(x'). The solution is very easily obtained from the
integral (1.17) for the potential:
J x — x 
with expansion (3.70) for l/x  x' . Since we are interested at the
moment in the potential outside the charge distribution, r < = r' and
r. Then we find:
d)(x) = 4^2 iy^j* 7 ** ( 6 '> <£»( x ') d * x '_
l,m
Y lm (6, <f>)
„i+i
(4.2)
(4.3)
Consequently the coefficients in (4.1) are:
qim=\Y l * m (d',cf>'y l p(x')d*x'
These coefficients are called multipole moments. To see the physical inter
pretation of them we exhibit the first few explicitly in terms of cartesian
coordinates :
4oo
= i JW)**'^,
4n
Qio =
=  fL \{x'  iy') P (x') d z x' =  /i (p x  i Py )
N 877 J 'V 077
jz'pV
)d 3 x>=/±p
Att
(4.4)
(4.5)
(4.6)
AN 2ttJ ll'S Ztt
Only the moments with m > have been given, since (3.54) shows that for
a real charge density the moments with m < are related through
«i.» = (D m «
im
(4.7)
100
Classical Electrodynamics
In equations (4.4)(4.6), q is the total charge, or monopole moment, p is
the electric dipole moment :
p = j x>(x') dV
and Q tj is the quadrupole moment tensor :
<2„ = j(3x t 'z/  r'%^p{x') d*x'
(4.8)
(4.9)
We see that the /th multipole coefficients [(21 + 1) in number] are linear
combinations of the corresponding multipoles expressed in rectangular
coordinates. The expansion of <D(x) directly in rectangular coordinates :
" p • X , 1 "'O _ X.Xi
O(x) = ^ +
r
by direct Taylor's series expansion of l/x — x' will be left as an exercise
for the reader. It becomes increasingly cumbersome to continue the
expansion in (4.10) beyond the quadrupole terms.
The electric field components for a given multipole can be expressed
most easily in terms of spherical coordinates. The negative gradient of a
term in (4.1) with definite /, m has spherical components:
= Mi±j) Y lm (d, <f>)
21 + 1 qim r l+2
E e = 
47T
E^= 
21+ 1
47T
21+ 1
1 r)
lim j^i rr Y lm (d, (f>)
dd
111
im
sin
Y lm(0, <f>)
(4.11)
d Y lm ldQ and Y lm {sm d can be expressed as linear combinations of other
F Jm 's, but the expressions are not particularly illuminating and so will be
omitted. The proper way to describe a vector multipole field is by vector
spherical harmonics, discussed in Chapter 16.
For a dipole p along the z axis, the fields in (4.11) reduce to the familiar
form:
2p cos 6
„3
£„ =
E a =
r
p sin 6
E^ =
(4.12)
These dipole fields can be written in vector form by recombining (4.12) or
by directly operating with the gradient on the dipole term in (4.10). The
[Sect. 4.2] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 101
result for the field at a point x due to a dipole p at the point x' is:
3n(p • n)  p
x  xf
where n is a unit vector directed from x' to x.
E(x) = J ur"'* (4.13)
Ix  x' 3
4.2 Multipole Expansion of the Energy of a Charge Distribution in an
External Field
If a localized charge distribution described by />(x) is placed in an
external potential O(x), the electrostatic energy of the system is:
W
= fp(x)0(x) d 3 x (4.14)
If the potential O is slowly varying over the region where p(x) is non
negligible, then it can be expanded in a Taylor's series around a suitably
chosen origin:
O(x) = O(0) + x • VO(0) + \ S Sx iXj ^~ (0) + • • • (4.15)
2, ' t ' ( VX* OX j
i )
Utilizing the definition of the electric field E =  VO, the last two terms
can be rewritten. Then (4.15) becomes:
O(x) = O(0)  x • E(0)  i 22*^' Ifr (0) + " '
% i
Since V • E = for the external field, we can subtract
ir 2 V • E(0)
from the last term to obtain finally the expansion :
o(x) = o(0)  x . e(0)  i 22 (3a: ^ " r2di3) iv (0) + " ' (4 ' 16)
t 3
When this is inserted into (4.14) and the definitions of total charge, dipole
moment (4.8) and quadrupole moment (4.9), are employed, the energy
takes the form:
W = <?O(0)  P • E(0)  i y V(2 W ^ (0) + • • • (4.17)
6 *—i *—* ox,
i i l
This expansion shows the characteristic way in which the various multi
poles interact with an external field — the charge with the potential, the
dipole with the electric field, the quadrupole with the field gradient, and
so on.
102 Classical Electrodynamics
In nuclear physics the quadrupole interaction is of particular interest.
Atomic nuclei can possess electric quadrupole moments, and their magni
tudes and signs have a bearing on the forces between neutrons and protons,
as well as the shapes of the nuclei themselves. The energy levels or states
of a nucleus are described by the quantum numbers of total angular
momentum / and its projection M along the z axis, as well as others which
we will denote by a general index a. A given nuclear state has associated
with it a quantummechanical charge density* p JM <k*)> which depends
on the quantum numbers (/, M, a), but which is cylindrically symmetric
about the z axis. Thus the only nonvanishing quadrupole moment is q 20
in (4.6), or Q 33 in (4.9).f The quadrupole moment of a nuclear state is
defined as the value of (l/e) Q^ with the charge density p JMx ( x )> where e
is the protonic charge :
 1 .!**
Qj Ma = " J (3^  r') PjMa (x) d*x (4.18)
The dimensions of Qjm<x are consequently (length) 2 . Unless the circum
stances are exceptional (e.g., nuclei in atoms with completely closed
electronic shells), nuclei are subjected to internal fields which possess field
gradients in the neighborhood of the nuclei. Consequently, according to
(4. 17), the energy of the nuclei will have a contribution from the quadrupole
interaction. The states of different M value for the same J will have
different quadrupole moments Q JM(X , and so a degeneracy in M value
which may have existed will be removed by the quadrupole coupling to the
"external" (crystal lattice, or molecular) electric field. Detection of these
small energy differences by radiofrequency techniques allows the deter
mination of the quadrupole moment of the nucleus. {
The interaction energy between two dipoles p x and p 2 can be obtained
directly from (4.17) by using the dipole field (4.13). Thus, the mutual
potential energy is
w = P 1 P23(n.p 1 )(n.p 2 )
l*i  x 2  3
where n is a unit vector in the direction (x x — x 2 ). The dipoledipole
interaction is attractive or repulsive, depending on the orientation of the
dipoles. For fixed orientation and separation of the dipoles, the value of
* See Blatt and Weisskopf, pp. 23 ff., for an elementary discussion of the quantum
aspects of the problem.
t Actually Q xx and Q 22 are different from zero, but are not independent of Q 33 , being
given by e„ = Q 22 = —%Q 33 .
+"The quadrupole moment of a nucleus," denoted by Q, is defined as the value of
Q JMa in the state M = J. See Blatt and Weisskopf, he. cit.
[Sect. 4.3] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 103
the interaction, averaged over the relative positions of the dipoles, is zero.
If the moments are generally parallel, attraction (repulsion) occurs when
the moments are oriented more or less parallel (perpendicular) to the line
joining their centers. For antiparallel moments the reverse is true. The
extreme values of the potential energy are equal in magnitude.
4.3 Macroscopic Electrostatics; Effects of Aggregates of Atoms
The equations
V • e = Atto )
(4.20)
V x e = 0j
govern electrostatic phenomena of all types, provided the "microscopic"
electric field € is derived from the total "microscopic" charge density p.
For problems with a few idealized point charges in the vicinity of mathe
matically defined boundary surfaces, equations (4.20) are quite acceptable.
But there are many physical situations in which a complete specification
of the problem in terms of individual charges would be impossible. Any
problem involving fields in the presence of matter is a case in point. A
macroscopic amount of matter has of the order of 10 23±3 charges in it, all
of them in motion to a greater or lesser extent because of thermal agitation
or zero point vibration.
Setting aside the question of whether electrostatics can be relevant to a
situation in which the charges are in incessant motion, let us consider the
task of handling macroscopic problems with large numbers of atoms or
molecules. Clearly the solution for the electric field :
<x) = \GLZ*l pW fitf (4.21)
J x — x  d
is not very suitable, since (a) it involves a charge density p which must
specify the exact positions of very many charges, and (b) it fluctuates
wildly as the observation point moves by only very small distances (of the
order of atomic dimensions). Fortunately, for macroscopic electrostatics
we do not want as detailed information as is contained in (4.21). We are
content with averages of electric field strengths over regions of the order
of 10~ 6 cm 3 (i.e., 10 2 cm linear dimension) or greater. Since atomic
volumes are of the order of 10~ 24 cm 3 , there are of the order of 10 18 or more
atoms in the volumes of macroscopic interest. This means that the micro
scopic fluctuations will be entirely averaged out. We will wish to deal with
an average c(x) and p'(x). The averages will be over a macroscopically
104 Classical Electrodynamics
small volume AV, large enough, however, to contain very many atoms or
molecules : _
<e(x)> = L e(x + I) d^
AV J A
' AV
AVJav ■— J
(4.22)
The averaged quantities are denoted by angle brackets ( ) ; the variable
\ ranges over the volume A V.
The averaging procedure now allows us to answer the question of
whether it is legitimate to talk in static terms when the charges in matter
are in thermal motion. At any instant of time the very many charges in
the volume A V will be in all possible states of motion. An average over
them at that instant will yield the same result as an average at some later
instant of time. Hence, as far as the averaged quantities are concerned,
it is legitimate to talk of static fields and charges.* Furthermore, the
averaging can be done as if the atomic charges were fixed in space at the
positions they have at some arbitrary instant. Hence the situation can be
regarded as electrostatic even at the microscopic level for purposes of
calculation.
In the treatment of macroscopic electrostatics it is useful to break up
the averaged charge density (p(x)) into two parts, one of which is the
averaged charge of the atomic or molecular ions, or excess free charge
placed in or on the macroscopic body, and the other of which is the
induced or polarization charge. In the absence of external fields, atoms
or molecules may or may not have electric dipole moments, but if they do,
the moments are randomly oriented. In the presence of a field, the atoms
become polarized (or their permanent moments tend to align with the
field) and possess on the average a dipole moment These dipole moments
can contribute to the averaged charge density (/>'( x )) Since the induced
dipole moments tend to be proportional to the applied field, we will find
that the macroscopic version of (4.20) will involve only one constant to
characterize the average polarizability of the medium involved.
To see how the induced dipole moments enter the problem we first
consider the microscopic field due to one molecule with center of mass at
the point x, in Fig. 4. 1 while the observation point is at x. The molecular
charge density is p/(x'), where x' is measured from the center of mass of
the molecule. It should be noted that p in general depends on the position
of Xj of the molecule, since the distortion of the charge cloud depends on
the local field present. The microscopic electric field due to the y'th
* This ignores the very small (at room temperature) induction and radiation fields due
to the acceleration of the charges in their thermal motion.
[Sect. 4.3] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 105
Fig. 4.1 A molecule with center of
mass at x s gives a contribution to
the potential at the point P with
position x. The internal coordinate
x' is measured from the center of
mass.
Molecule
molecule is
e,(x)
Jmol
/>/(*')
x — x, — x
dV
(4.23)
For observation points outside the molecule we can expand in multipoles
around the center of mass of the molecule. According to (4.10), this leads
to r / i
e J— + v/ 1
6,(X) = V
Lx  x,
(^i)—]
(4.24)
where
e. = \ P /(x') d*x'
Pi = f x' Pj '(x')d*x
(4.25)
are the molecular charge and the dipole moment, respectively. The
quadrupole term in (4.10) could have been retained, but as long as the
macroscopic variations of field occur over distances large compared to
molecular dimensions it contributes negligibly to the averaged field
relative to the dipole term. Both e j and p 3 are functions of the position of
the molecule.
To obtain the microscopic field due to all the molecules we sum over;:
€(X)
=  v 2[— — + p> * ^.(i — ^ — r)l ( 4  26 >
f*Lxx, \xx,/J
We now want to average according to (4.22) in order to obtain a macro
scopic field. To facilitate this averaging procedure we replace the discrete
sum over the molecules by an integral by introducing apparently con
tinuous charge and polarization densities :
Pmol(x) = 2 e A X  X >)
i
Wmol(x) = 2 P,<5(X  X,)
(4.27)
106 Classical Electrodynamics
Then (4.26) can be rewritten formally as :
:(x) = V (d 3 x"\^^l + „ mol (x") • V"( 1 —)~
J Lxx" \xx"/J
(4.28)
To illustrate the averaging process we consider the first term in (4.28).
The averaged value is, by (4.22):
< €l (x)> = V
^ f d 3 z(d 3 x"
.AV Jav J
pmoljx")
x + 5
X"U
(4.29)
where we have used the fact that differentiation and averaging can be
interchanged. If the variable of integration x" is replaced by x" = x' + %,
then
< Cl (x)> = V L d^ \d 3 x> pmol(x ' + 5) 1 ( 4.30)
LAV Jav J x  x' J
The equality of (4.29) and (4.30) shows the obvious equivalence of averag
ing by means of moving the observation point around the volume AV
centered at x and averaging by moving the integration point over the
molecules in a volume A V centered around x'. From definition (4.27) it
is clear that the integral of p mol over the volume AF at x' just adds up all
the molecular charges e, inside A V:
±J WpmolV + 5) = i J
AVJav ^V^
If the macroscopic density of molecules at x' is N(x') molecules per unit
volume and (e mol (x')> is the average charge per molecule within the volume
AFatx', then
~ I ^ 3 ^moi(x' + J) = N(x')<e mol (x')> (4.31)
LA V J AV
Now (4.30) can be written
(6 1 (x)) = vf iV(x/)(gm0l(X,) >^
J x  x'l
Exactly similar considerations can be made for the second term in (4.28).
With the same definitions of averages we have
e.
j I d*£n mol (x' + ?) = iV(x')<p m oi(x')>
(4.32)
Then the averaged form of (4.28) is given by:
<€(x)> = V (N(x'){ ( * mol(x y + <p mo i(x')> • V'(— I )) dV (4.33)
J \ x — x I \x — x'\/)
[Sect. 4.3] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 107
To obtain the macroscopic equivalent of (4.20) we take the divergence of
both sides. Recalling that V 2 (l/x  x') = 4tt6(x  x'), we find:
V • <€(x)> = 4tt fiV(x'){<emoi(x')><5(x  x') + <Pmoi(x')> • V'<3(x  x')} dV
From the properties of the delta function (Section 1.2) it follows that
V • <€(x)> = 47rJV(x)<e mol (x)>  4ttV • (JV(x)<p mol (x)» (4.34)
This is of the form of the first equation of (4.20) with the charge density />'
replaced by two terms, the first being the average charge per unit volume
of the molecules and the second being the polarization charge per unit
volume. The presence of the divergence in the polarizationcharge density
seems very natural when one thinks of how this part of the charge density is
created. If we consider a small volume in the medium, part of the charge
inside that volume may be due to the net charges on the molecules. But
there is a contribution arising from the polarization of the charge cloud of
the molecules in an external field, since, for example, molecules whose
charge once lay totally inside the volume may now have part of their
charge cloud outside the volume in question. If the polarization is uniform
over the space containing our small volume, then as much charge will be
brought in through the surface of the volume as will leave it, and there will
be no net effect. But if the polarization is not uniform, there can be a net
increase or decrease of charge within the volume, as indicated schemati
cally in Fig. 4.2. This is the physical origin of the polarizationcharge
density.
In (4.34) the two divergences can be combined so that the equation
V • [(€> + 477iV(p mol >] = 47rN(e mol ) (4.35)
It is customary to introduce certain macroscopic quantities, namely, the
electric field E, the polarization P (electric dipole moment per unit volume),
Fig. 4.2 Origin of polarizationcharge density.
Because of spatial variation of polarization more
molecular charge may leave a given small volume
than enters it.
108 Classical Electrodynamics
the charge density p, and the displacement D, defined as follows :
E = <€>
P = ^<Pmol>
P = N{e mol )
D = E + 4ttP
If there are several different kinds of atoms or molecules in the medium
and perhaps extra charge is added, these definitions have the obvious
generalizations : _
p = 2>«<p«>
(4.36)
P = INM + Pe
(4.37)
where N { is the number of molecules of type * per unit volume, (e t ) is their
average charge, and (p^) is their average dipole moment. p ex is the excess
(or free) charge density. Usually the molecules are neutral, and the total
charge density p is just the free charge density.
With the definitions of (4.36) or (4.37), the macroscopic divergence
equation becomes : „ _. . .,,„,
n V • D = 4tt/> (4.38)
The macroscopic equivalent of the other member of the pair (4.20) can be
obtained by taking the curl of (4.33). Obviously the result is
V x E = (4.39)
For macroscopic electrostatic problems in the presence of dielectrics,
(4.38) and (4.39) replace the microscopic equations (4.20).
The solution for the electric field (4.33) can be expressed in terms of the
macroscopic variables as
E(x)
= V [d z x'\ p(x,) + P(x') • V'( )
J Llxx'l \xx'/J
(4.40)
The second term, describing the dipole field, has already been discussed in
Section 1.6.
4.4 Simple Dielectrics and Boundary Conditions
It was mentioned in the previous section that the molecular polarization
depends on the local electric field at the molecule. In the absence of a
field there is no average polarization.* This means that the polarization
* Except for electrets, which have a permanent electric polarization.
[Sect. 4.4] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 109
P, which is in general a function of E, can be expanded as a powers series
in the field, at least for small fields. Any component will have an expansion
of the form: _, ~ „ , ^, F ^ .
Pi = 2<*h e j + 2 b m E > E k + * * '
A priori it is not clear how important the higher terms will be in practice.
Experimentally it is found that the polarization as a function of applied
field looks qualitatively as shown in Fig. 4.3. At normal temperatures and
for fields attainable in the laboratory the linear approximation is completely
adequate. This is not surprising if it is remembered that interatomic
electric fields are of the order of 10 9 volts/cm. Any external field causing
polarization is only a small perturbation. For a general anisotropic
medium (e.g., certain crystals such as calcite and quartz), there can be six
independent elements a u . But for simple substances, called isotropic, P
is parallel to E with a constant of proportionality % e which is independent
of the direction of E. Then p _ g (4.41)
The constant % e is called the electric susceptibility of the medium. We
then find the displacement proportional to E:
D = eE (4.42)
where
€ = 1 + 47T Xe (4.43)
is the dielectric constant.
If the dielectric is not only isotropic, but also uniform, e is independent
of position. Then the divergence equation can be written
VE = — p
€
(4.44)
and all problems in that medium are reduced to those of previous chapters,
except that the electric fields produced by given charges are reduced by a
Fig. 4.3 Components of polariza
tion as a function of applied
electric field.
110 Classical Electrodynamics
Region 2
Region 1
Fig. 4.4
factor 1/e. The reduction can be understood in terms of a polarization of
the atoms which produce fields in opposition to that of the given charge.
One immediate consequence is that the capacitance of a capacitor is increased
by a factor of e if the empty space between the electrodes is filled with a di
electric with dielectric constant e (true only to the extent that fringing fields
can be neglected).
An important consideration is the boundary conditions on the field
quantities E and D at surfaces where the dielectric properties vary dis
continuously. Consider a surface S as shown in Fig. 4.4. The unit vector
n is normal to the surface and points from region 1 with dielectric constant
e 1 to region 2 with dielectric constant e 2 . In exactly the same manner as in
Section 1.6 we find, by taking a Gaussian pill box with end faces in regions
1 and 2 parallel to the surface 5*, that
(D 2  DO • n = 4tt<7 (4.45)
where a is the surfacecharge density (not including polarization charge).
Similarly, by applying Stokes's theorem to V x E = 0, we find that
(E x  E 2 ) x n = (4.46)
These boundary conditions on the normal component of D and the
tangential component of E replace the microscopic conditions (1.22) and
below. The macroscopic equivalent of (1.22) can be recovered from (4.45)
by extracting the polarizationcharge density from the lefthand side.
4.5 BoundaryValue Problems with Dielectrics
The methods of previous chapters for the solution of electrostatic
boundary value problems can readily be extended to handle the presence
of dielectrics. In this section we will treat a few examples of the various
techniques applied to dielectric media.
[Sect. 4.5] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 111
To illustrate the method of images for dielectrics we consider a point
charge q embedded in a semiinfinite dielectric €l a distance d away from a
plane interface which separates the first medium from another semiinfinite
dielectric e 2 . The surface may be taken as the plane z = 0, as shown in
Fig. 4.5. We must find the appropriate solution to the equations:
€l V • E = 4irp, z >
€2 V.E = 0, *<0 I (447)
and V x E = 0, everywhere
subject to the boundary conditions at z = :
[ E y J I E b
Since V x E = everywhere, E is derivable in the usual way from a
potential <D. In attempting to use the image method it is natural to locate
an image charge q' at the symmetrical position A' shown in Fig. 4.6. Then
for z > the potential at a point P described by cylindrical coordinates
( P , <f>, z) will be \ (a a'\ <a „m
lim
(4.48)
where R x = V P 2 + (d  *) 2 , U, = V P * + (d + zf. So far the pro
cedure is completely analogous to the problem with a conducting material
in place of the dielectric e 2 for z < 0. But we now must specify the potential
for z < 0. Since there are no charges in the region z < 0, it must be a
solution of Laplace's equation without singularities in that region. Clearly
the simplest assumption is that for z < the potential is equivalent to that
of a charge q" at the position A of the actual charge q\
O =  —
z <0
(4.50)
Fig. 4.5
112
Classical Electrodynamics
62
p
Ry^
^^ VRi
q *^^^
\ 9
A '!c d >
d >\ A
i
i
Fig. 4.6
Since
while
1(1
1(1)
_ d(M
d
z = dz\#
Ht)
2^
_ — P
z =
o~ (p 2 + d 2 f A
the boundary conditions (4.48) lead to the requirements :
q  q' = q"
(q+q') = q"
e l e 2
These can be solved to yield the image charges q' and q" :
q' = ~
+ «i
«
(4.51)
For the two cases e 2 > e x and e 2 < e x the lines of force are shown qualita
tively in Fig. 4.7.
The polarizationcharge density is given by — V«P. Inside either
dielectric, P = # e E, so that — V • P = — # e V • E = 0, except at the point
charge q. At the surface, however, % e takes a discontinuous jump,
Ax e = (l/47r)(ei — e 2 ) as z passes through z = 0. This implies that there
is a polarization surfacecharge density on the plane 2 = 0:
<Vi=(P 2 Pi)n (4.52)
where n is the unit normal from dielectric 1 to dielectric 2, and P 4 is the
polarization in the dielectric / at z = 0. Since
P,=
4n
Vte 1 )™
[Sect. 4.5] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 113
€2>ei
«2<«1
Fig. 4.7 Lines of electric force for a point charge embedded in a dielectric e x near a semi
infinite slab of dielectric e 2 .
it is a simple matter to show that the polarizationcharge density is
CTpol =
q in — € i) d
27re 1 (e 2 + e 1 )0> 2 + d 2 ) 5
(4.53)
In the limit e 2 > e x the dielectric e 2 behaves much like a conductor in that
the field inside it becomes very small and the surfacecharge density (4.53)
approaches the value appropriate to a conducting surface.
The second illustration of electrostatic problems involving dielectrics is
that of a dielectric sphere of radius a with dielectric constant e placed in an
initially uniform electric field which at large distances from the sphere is
directed along the z axis and has magnitude E , as indicated in Fig. 4.8.
Both inside and outside the sphere there are no free charges. Consequently
the problem is one of solving Laplace's equation with the proper boundary
conditions at r = a. From the axial symmetry of the geometry we can
Fig. 4.8
H4 Classical Electrodynamics
take the solution to be of the form:
inside:
®in = lA l rip i (cose)
1=0
outside: OTt = 2 [B t r l + Cy^+^cos 0)
1 =
(4.54)
(4.55)
From the boundary condition at infinity (O > — E z = —E r cos 0) we
find that the only nonvanishing B % is B ± = E . The other coefficients are
determined from the boundary conditions at r = a:
1 dO
tangential E:
NORMAL D:
a dd
1 ao
out
dr
a dd
50 mit
dr
(4.56)
The first boundary condition leads to the relations :
A x = E + ^
a 6
1 „2l+l>
for / ^ 1
(4.57)
while the second gives :
J
^ 1= £ 2^
el A, = (/ + 1)
C,
for / ^ 1
(4.58)
The second equations in (4.57) and (4.58) can be satisfied simultaneously
only with A l = C t = for all / ^ 1. The remaining coefficients are given
in terms of the applied electric field E :
(4.59)
The potential is therefore
<E>in =  \—^—)E r cos 6
O ut = — E r cos d +
€ + 2
2/*° r 2
cos
(4.60)
[Sect. 4.5] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 115
The potential inside the sphere describes a constant electric field
parallel to the applied field with magnitude
£m = — , E < E (4.61)
e + 2
Outside the sphere the potential is equivalent to the applied field E plus
the field of an electric dipole at the origin with dipole moment:
■ fcriW
(4.62)
oriented in the direction of the applied field. The dipole moment can be
interpreted as the volume integral of the polarization P. The polarization
is
P= (i^i) E = l(i^i) Eo ( 4 .63)
\ 477 / 47T\€ + 2/
It is constant throughout the volume of the sphere and has a volume
integral given by (4.62). The polarization surfacecharge density is,
according to (4.52), <r pol = (P • r)/r:
Opol
lrr\7V~2) E0
cos (4.64)
This can be thought of as producing an internal field directed oppositely
to the applied field, so reducing the field inside the sphere to its value (4.61),
as sketched in Fig. 4.9.
The problem of a spherical cavity of radius a in a dielectric medium with
dielectric constant e and with an applied electric field E parallel to the z
axis, as shown in Fig. 4.10, can be handled in exactly the same way as the
dielectric sphere. In fact, inspection of boundary conditions (4.56) shows
that the results for the cavity can be obtained from those of the sphere by
the replacement e > (1/e). Thus, for example, the field inside the cavity
E
Eq
,< /+
Fig. 4.9 Dielectric sphere in a uniform field E , showing the polarization on the left
and the polarization charge with its associated, opposing, electric field on the right.
116
Classical Electrodynamics
Fig. 4.10. Spherical cavity in a
dielectric with a uniform field
applied.
is uniform, parallel to E , and of magnitude :
3e
E\n =
2e+ 1
E > E
(4.65)
Similarly, the field outside is the applied field plus that of a dipole at the
origin oriented oppositely to the applied field and with dipole moment:
'■ferrK
(4.66)
4.6 Molecular Polarizability and Electric Susceptibility
In this section and the next we will consider the relation between
molecular properties and the macroscopically defined parameter, the
electric susceptibility Xe  Our discussion will be in terms of simple
classical models of the molecular properties, although a proper treatment
necessarily would involve quantummechanical considerations. Fortu
nately, the simpler properties of dielectrics are amenable to classical
analysis.
Before examining how the detailed properties of the molecules are related
to the susceptibility we must make a distinction between the fields acting
on the molecules in the medium and the external field. The susceptibility
is defined through the relation P = Xe E, where E is the macroscopic
electric field. In rarefied media where molecular separations are large
there is little difference between the macroscopic field and that acting on
any molecule or group of molecules. But in dense media with closely
packed molecules the polarization of neighboring molecules gives rise to
an internal field E, at any given molecule in addition to the average
macroscopic field E, so that the total field at the molecule is E + E,. The
internal field can be written as
E,
(t+>
(4.67)
[Sect. 4.6] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 117
where sF is the contribution of molecules close to the given molecule, and
(4tt/3)P is the contribution of the more distant molecules. It is customary
to consider the two parts separately by imagining a spherical surface of size
large microscopically but small macroscopically surrounding a molecule,
as shown in Fig. 4.11, and determining the field at the center due to the
polarization of the molecules exterior to the sphere and the resulting charge
density induced on the surface of the sphere. This charge density is
— P . n, where n is the outward normal from the spherical surface. The
resulting field at the center is obviously parallel to P and has the magnitude :
E a> f r2 da (^ cos excos o) = to p (4 68)
J sphere T *
giving the first term in (4.67).
The field sP due to the molecules near by is more difficult to determine.
Lorentz (p. 138) showed that for atoms in a simple cubic lattice s = at
any lattice site. The argument depends on the symmetry of the problem,
as can be seen as follows. Suppose that inside the sphere we have a cubic
array of dipoles such as are shown in Fig. 4.12, with all their moments
constant in magnitude and oriented along the same direction (remember
that the sphere is macroscopically small). The positions of the dipoles are
given by the coordinates x ijk with the components along the coordinate
axes {ia, ja, ka), where a is the lattice spacing, and i,j, k each take on
positive and negative integer values. The field at the origin due to all the
dipoles is, according to (4.13),
E =
2 3(P • xnk) x ijk — x ijkV (4.69)
i,i,k
x ijk
The x component of the field can be written in the form : :
P ^ 3(i 2 Pl + Up* + ikp z )  (i 2 + f + fc 2 )Pi (470)
ilk
Spherical ^""" ~~^ N s\ti
surface ~^V V
Fig. 4.11 Calculation of the internal field —
contribution from distant molecules.
I — » — 
I ^ I p
\ Molecule ,'
V V ^
118
Classical Electrodynamics
Fig. 4.12 Calculation of the in
ternal field — contribution from near
by molecules in a simple cubic lattice.
Since the indices run equally over positive and negative values, the cross
terms involving (ifp 2 + ikp 3 ) vanish. By symmetry the sums:
y i2 V f y k 2
£ (* 2 + f + k*t £ (, + f + k*f* £ 0* + f + k*t
are all equal. Consequently
E = V [3i 2 Q 2 + j 2 + /c 2 )] Pl _
1 / . a, .9. . .9. . , o^te — "
it «V +/ + **)
2\^
(4.71)
ijk
Similar arguments show that the y and z components vanish also. Hence
s = for a simple cubic lattice.
If s = for a highly symmetric situation, it seems plausible that s =
also for completely random situations. Hence we expect amorphous
substances like glass to have no internal field due to nearby molecules.
Although calculations taking into account the structural details of the
substance are necessary to obtain an accurate answer, it is a good working
assumption that s ~ for almost all materials.
The polarization vector P was defined in (4.36) as
P = ^<Pmol>
where (p mol > is the average dipole moment of the molecules. This dipole
moment is approximately proportional to the electric field acting on the
molecule. To exhibit this dependence on electric field we define the mole
cular polarizability y mol as the ratio of the average molecular dipole
moment to the applied field at the molecule. Taking account of the internal
field (4.67), this gives :
<Pmol> = 7mol(E + E,) (4.72)
>mol
is, in principle, a function of the electric field, but for a wide range of
[Sect. 4.7] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 119
field strengths is a constant which characterizes the response of the
molecules to an applied field (see Section 4.4). Equation (4.72) can be
combined with (4.36) and (4.67) to yield:
P = N)w(E + yP) (4.73)
where we have assumed 5 = 0. Solving for P in terms of E and using the
fact that P = % e E defines the electric susceptibility of a substance, we find
X. = N I m ° l ( 4  74 >
47T
1 Ny m0 \
3
as the relation between susceptibility (the macroscopic parameter) and
molecular polarizability (the microscopic parameter). Since the dielectric
constant is e = 1 + 4tt^ 6 , it can be expressed in terms of y mol , or
alternatively the molecular polarizability can be expressed in terms of the
dielectric constant :
7mol = ^(^) (4.75)
4ttN\€ + 2/
This is called the ClausiusMossotti equation, since Mossotti (in 1850) and
Clausius independently (in 1879) established that for any given substance
( e _ i)/( e + 2) should be proportional to the density of the substance.*
The relation holds best for dilute substances such as gases. For liquids
and solids, (4.75) is only approximately valid, especially if the dielectric
constant is large. The interested reader can refer to the books by Bottcher,
Debye, and Frohlich for further details.
4.7 Models for the Molecular Polarizability
The polarization of a collection of atoms or molecules arises in two ways :
(a) the applied field distorts the charge distributions and so produces
an induced dipole moment in each molecule ;
(b) the applied field tends to line up the initially randomly oriented
permanent dipole moments of the molecules.
To estimate the induced moments we will consider a simple model of
* At optical frequencies, e = « 2 , where n is the index of refraction. With n 2 replacing
e in (4.75), the equation is sometimes called the LorentzLorenz equation (1880).
120 Classical Electrodynamics
harmonically bound charges (electrons). Each charge e is bound under
the action of a restoring force
F = ma) 2 x (4.76)
where m is the mass of the charge, and o the frequency of oscillation
about equilibrium. Under the action of an electric field E the charge is
displaced from its equilibrium by an amount x given by
mco Q 2 x = eE
Consequently the induced dipole moment is
e 2
Pmoi = ex = E (4.77)
ma>.
This means that the polarizability is y = e 2 {mco 2 . If there are Z electrons
per molecule,^ having a restoring force constant mco? QT/} = Z), then the
molecular polarizability due to the electrons is : j
Y*= L y— % (4.78)
i
To get a feeling for the order of magnitude of y el we can make two
different estimates. Since y has the dimensions of a volume, its magnitude
must be of the order of molecular dimensions or less, namely y el < 10~ 23
cm 3 . Alternatively, we note that the binding frequencies of electrons in
atoms must be of the order of light frequencies. Taking a typical wave
length of light as 3000 angstroms, we find co ~ 6 x 10 15 sec 1 . Then
Yei ^ (e 2 /ma> 2 ) ~6x 10~ 24 cm 3 , consistent with the molecular volume
estimate. For gases at NTP the number of molecules per cubic centimeter
is TV = 2.7 x 10 19 , so that their susceptibilities should be of the order of
X e < 10 4 . This means dielectric constants differing from unity by a few
parts in 10 3 , or less. Experimentally, typical values of dielectric constant
are 1.00054 for air, 1.0072 for ammonia vapor, 1.0057 for methyl alcohol,
1.000068 for helium. For solid or liquid dielectrics, N~ 10 22  10 23
molecules/cm 3 . Consequently, the susceptibility can be of the order of
unity (to within a factor 10 ±r ) as is observed.*
The possibility that thermal agitation of the molecules could modify the
result (4.78) for the induced dipole polarizability needs consideration. In
statistical mechanics the probability distribution of particles in phase
* See Handbook of Chemistry and Physics, Chemical Rubber Publishing Co., or
American Institute of Physics Handbook, McGrawHill, New York, (1957), for tables of
dielectric constants of various substances.
[Sect. 4.7] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 111
space (p, q space) is proportional to the Boltzmann factor
expiH/kT) (4.79)
where H is the Hamiltonian. In the simple problem of a harmonically
bound electron with an applied field in the z direction, the Hamiltonian is
H = — p 2 +  eo 2 x 2  eEz (4.80)
2m 2
where here p is the momentum of the electron. The average value of the
dipole moment is
(d 3 p(d 3 x(ez)exp(HlkT)
<*w> = ^j (4.81)
d*p d*x exp (H/kT)
The integration over (d z p) and (dx dy) can be done immediately to yield
(pmol) =
e\dzz exp
r_ i /W z2 eE Y]
L kT\ 2 /J
dz exp
1 W 2 2 _ eE X
. kT\ 2 /J
An integration by parts in the numerator yields the result :
e 2
(pmoi) = ; E
ma) Q
the same as was found in (4.77) by elementary means, ignoring thermal
motion. Thus the molecular polarizability (4.78) holds even in the presence
of thermal motion.
The second type of polarizability is that caused by the partial orientation
of randomly oriented permanent dipole moments. This orientation polari
zation is important in "polar" substances such as HC1 and H 2 and was
first discussed by Debye (1912). All molecules are assumed to possess a
permanent dipole moment p which can be oriented in any direction in
space. In the absence of a field thermal agitation keeps the molecules
randomly oriented so that there is no net dipole moment. With an applied
field there is a tendency to line up along the field in the configuration of
lowest energy. Consequently there will be an average dipole moment. To
calculate this we note that the Hamiltonian of the molecule is given by
H = H  p • E (4.82)
122 Classical Electrodynamics
' Nonpolar
Fig. 4.13. Variation of molec
ular polarizability y m oi with
temperature for polar and non
polar substances. y m oi is plot
\IT ■>■ ted versus T _1 .
where H Q is a function of only the "internal" coordinates of the molecule.
Using the Boltzmann factor (4.79), we can write the average dipole
moment as :
[dnp cos6ttp(*£™l\
<Pmoi> = j— ^— (4.83)
where we have chosen E along the z axis, integrated out all the irrelevant
variables, and noted that only the component of p parallel to the field is
different from zero. In general, (p E/kT) is very small compared to unity,
except at low temperatures. Hence we can expand the exponentials and
obtain the result:
<Pna)^~ E ( 4  84 )
3 kT
We note that the orientation polarization depends inversely on the tempera
ture, as might be expected of an effect in which the applied field must
overcome the opposition of thermal agitation.
In general both types of polarization, induced (electronic) and orienta
tion, are present, and the general form of the molecular polarization is
ymoi ~ yei +  t (4.85)
3 kT
This shows a temperature dependence of the form (a + b/T) so that the
two types of polarization can be separated experimentally, as indicated in
Fig. 4.13. For "polar" molecules, such as HC1 and H 2 0, the observed
permanent dipole moments are of the order of an electronic charge times
10~ 8 cm, in accordance with molecular dimensions.
[Sect. 4.8] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 123
4.8 Electrostatic Energy in Dielectric Media
In Section 1.11 we discussed the energy of a system of charges in free
space. The result
^=iL(x)(D(x)d 3 z (4.86)
for the energy due to a charge density p(x) and a potential O(x) cannot in
general be taken over as it stands in our macroscopic description of
dielectric media. The reason becomes clear when we recall how (4.86) was
obtained. We thought of the final configuration of charge as being
created by assembling bit by bit the elemental charges, bringing each one
in from infinitely far away against the action of the then existing electric
field. The total work done was given by (4.86). With dielectric media
work is done not only to bring real (macroscopic) charge into position,
but also to produce a certain state of polarization in the medium. If p
and 3> in (4.86) represent macroscopic variables, it is certainly not evident
that (4.86) represents the total work, including that done on the dielectric.
In order to be general in our description of dielectrics we will not
initially make any assumptions about linearity, uniformity, etc., of the
response of a dielectric to an applied field. Rather, let us consider a small
change in the energy bW due to some sort of change bp in the charge
density /> existing in all space. The work done to accomplish this change
is
dW = (dp(x)<I>(x)d z x (4.87)
where O(x) is the potential due to the charge density p(x) already present.
Since V • D = Anp, we can relate the change dp to a change in the dis
placement of <5D :
<5 P = — V05D) (4.88)
477
Then the energy change bW can be cast into the form:
bW = — f E • dD d*x (4.89)
4tt J
where we have used E = — V® and have assumed that p(x) was a localized
charge distribution. The total electrostatic energy can now be written
down formally, at least, by allowing D to be brought from an initial value
D = to its final value D :
W=— (d 3 x( D EbD (4.90)
477 J JO
124 Classical Electrodynamics
If the medium is linear, then
E • (5D = i(5(E • D) (4.91)
and the total electrostatic energy is
877 J
W=— E.Dft (4.92)
877 J
This last result can be transformed into (4.86) by using E = — VO and
V • D = Anp, or by going back to (4.87) and assuming that p and O are
connected linearly. Thus we see that (4.86) is valid macroscopically only
if the behavior is linear. Otherwise the energy of a final configuration must
be calculated from (4.90) and might conceivably depend on the past
history of the system (hysteresis effects).
A problem of considerable interest is the change in energy when a
dielectric object is placed in an electric field whose sources are fixed.
Suppose that initially the eJectric field E due to a certain distribution of
charges p (x) exists in a medium of dielectric constant e which may be a
function of position. The initial electrostatic energy is
^ = ^ fE .D d 3
077 J
where D = e E . Then with the sources fixed in position a dielectric
object of volume V 1 is introduced into the field, changing the field from E
to E. The presence of the object can be described by a dielectric constant
e(x), which has the value e x inside V 1 and e outside V x . To avoid mathe
matical difficulties we can imagine e(x) to be a smoothly varying function
of position which falls rapidly but continuously from e x to e at the edge
of the volume V x . The energy now has the value
W x = — E • D d 3 x
Sir J
where D = eE. The difference in the energy can be written :
W=± f(E.DE .D )^
077 J
1 f
.93)
= ^ f(E.D D.E )^ + ^ f(E + E ) • (D  D ) d*x (4.
The second integral can be shown to vanish by the following argument.
Since V x (E + E ) = 0, we can write
E + E = V0>
[Sect. 4.8] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 125
Then the second integral becomes :
j = _ J_ f VO • (D  D ) d 3 x
877 J
Integration by parts transforms this into
I = — I OV • (D  D ) d 3 x =
since V • (D — D ) = because the source charge density /> (x) is assumed
unaltered by the insertion of the dielectric object. Consequently the energy
change is
W = — f(E • D  D • E ) d z x (4.94)
877 J
The integration appears to be over all space, but is actually only over the
volume V x of the object, since, outside V lt D = e E. Therefore we can
write
W =  — f (e,  € ft )E • E d 3 x (4.95)
= __L f ( ei _ €o )E.E ^
87T JVi
If the medium surrounding the dielectric body is free space, then e = 1.
Using the definition of polarization P, (4.95) can be expressed in the form:
W= f PE d 3 x (4.96)
where P is the polarization of the dielectric. This shows that the energy
density of a dielectric placed in a field E whose sources are fixed is given
by
w=PE (4.97)
This result is analogous to the dipole term in the energy (4.17) of a charge
distribution in an external field. The factor \ is due to the fact that (4.97)
represents the energy density of a polarizable dielectric in an external field,
rather than a permanent dipole. It is the same factor \ which appears in
(4.91).
Equations (4.95) and (4.96) show that a dielectric body will tend to
move towards regions of increasing field E provided e x > e . To calculate
the force acting we can imagine a small generalized displacement of the
body <5f . Then there will be a change in the energy bW. Since the charges
are held fixed, there is no external source of energy and the change in field
126 Classical Electrodynamics
energy must be compensated for by a change in the mechanical energy of
the body. This means that there is a force acting on the body :
where the subscript Q has been placed on the partial derivative to indicate
that the sources of the field are kept fixed.
In practical situations involving the motion of dielectrics the electric
fields are often produced by a configuration of electrodes held at fixed
potentials by connection to an external source such as a battery. As the
distribution of dielectric varies, charge will flow to or from the battery to
the electrodes in order to maintain the potentials constant. This means that
energy is being supplied from the external source, and it is of interest to
compare the energy supplied in that way with the energy change found
above for fixed sources of the field. We will treat only linear media so that
(4.86) is valid. It is sufficient to consider small changes in an already
existing configuration. From (4.86) it is evident that the change in energy
accompanying the changes dp(x) and d<D(x) in charge density and potential
is
6W=U ><5(D + #<fy] d 3 x (4.99)
Comparison with (4.87) shows that, if the dielectric properties are not
changed, the two terms in (4.99) are equal. If, however, the dielectric
properties are altered,
e(x) * e(x) + Se(x) (4.100)
the contributions in (4.99) are not necessarily the same. In fact, we have
just calculated the change in energy brought about by introducing a
dielectric body into an electric field whose sources were fixed (dp = 0).
The reason for this difference is the existence of the polarization charge.
The change in dielectric properties implied by (4. 100) can be thought of as a
change in the polarizationcharge density. If then (4.99) is interpreted as an
integral over both free and polarizationcharge densities (i.e., a micro
scopic equation), the two contributions are always equal. However, it is
often convenient to deal with macroscopic quantities. Then the equality
holds only if the dielectric properties are unchanged.
The process of altering the dielectric properties in some way (by moving
the dielectric bodies, by changing their susceptibilities, etc.) in the presence
of electrodes at fixed potentials can be viewed as taking place in two steps.
In the first step the electrodes are disconnected from the batteries and the
[Sect. 4.8] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 127
charges on them held fixed (dp = 0). With the change (4.100) in dielectric
properties, the energy change is
dW 1 = \\pb<S> 1 d z x (4.101)
where dd> 1 is the change in potential produced. This can be shown to
yield the result (4.95). In the second step the batteries are connected again
to the electrodes to restore their potentials to the original values. There
will be a flow of charge dp 2 from the batteries accompanying the change in
potential* M> 2 = — 6^. Therefore the energy change in the second step
is
dW 2 = \ f(/x50 2 + <t>dp 2 ) d 3 x = 26W 1 (4.102)
since the two contributions are equal. In the second step we find the
external sources changing the energy in the opposite sense and by twice
the amount of the initial step. Consequently the net change is
6W= Upd^dPx (4.103)
Symbolically
bW v = 6W Q (4.104)
where the subscript denotes the quantity held fixed. If a dielectric with
€ > 1 moves into a region of greater field strength, the energy increases
instead of decreases. For a generalized displacement d£ the mechanical
force acting is now
F * = + (fi (4105)
REFERENCES AND SUGGESTED READING
The derivation of the macroscopic equations of electrostatics by averaging over
aggregates of atoms is presented by
Rosenfeld, Chapter II,
Mason and Weaver, Chapter I, Part III,
Van Vleck, Chapter 1.
Rosenfeld also treats the classical electron theory of dielectrics. Van Vleck's book is
devoted to electric and magnetic susceptibilities. Specific works on electric polarization
phenomena are those of
Bottcher,
Debye,
Frohlich.
* Note that it is necessary merely to know that M> 2 = — <5®i on the electrodes, since
that is the only place where free charge resides.
128
Classical Electrodynamics
Boundaryvalue problems with dielectrics are discussed in all the references on
electrostatics in Chapters 2 and 3.
Our treatment of forces and energy with dielectric media is brief. More extensive
discussions, including forces on liquid and solid dielectrics, the electric stress tensor,
electrostriction, and thermodynamic effects, may be found in
Abraham and Becker, Band 1, Chapter V,
Durand, Chapters VI and VII,
Landau and Lifshitz, Electrodynamics of Continuous Media,
Maxwell, Vol. 1, Chapter V,
Panofsky and Phillips, Chapter 6,
Stratton, Chapter II.
PROBLEMS
4.1 Calculate the multipole moments q lm of the charge distributions shown
below. Try to obtain results for the nonvanishing moments valid for all /, but
in each case find the first two sets of nonvanishing moments at the very least.
(a)
Total
charge
<?
Conducting circular
disc of radius
{d) For the charge distribution (Jo) write down the multipole expansion
for the potential. Keeping only the lowestorder term in the expansion, plot
the potential in the xy plane as a function of distance from the origin for
distances greater than a.
(e) Calculate directly from Coulomb's law the exact potential for (Jb) in the
xy plane. Plot it as a function of distance and compare with the result found
in (d).
Divide out the asymptotic form in parts (d) and (e) in order to see the
behavior at large distances more clearly.
4.2 A nucleus with quadrupole moment Q finds itself in a cylindrically symmetric
electric field with a gradient (dEjdz) along the z axis at the position of the
nucleus.
(a) Show that the energy of quadrupole interaction is
—HZ).
(b) If it is known that Q = 2 x 10~ 24 cm 2 and that WJh is 10 Mc/sec,
where h is Planck's constant, calculate (dE z /dz) in units of e/a 3 , where
a = h 2 /me 2 = 0.529 x 10~ 8 cm is the Bohr radius in hydrogen.
[Probs. 4] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 129
(c) Nuclearcharge distributions can be approximated by a constant
charge density throughout a spheroidal volume of semimajor axis a and
semiminor axis b. Calculate the quadrupole moment of such a nucleus,
assuming that the total charge is Ze. Given that Eu 153 (Z = 63) has a
quadrupole moment Q = 2.5 x 10 24 cm 2 and a mean radius
R = (a + b)\2 = 7 x 10 13 cm,
determine the fractional difference in radius (a — b)lR.
4.3 A localized distribution of charge has a charge density
/>(r) = Jr r 2 e~ r sin 2
647T
(a) Make a multipole expansion of the potential due to this charge
density and determine all the nonvanishing multipole moments. Write
down the potential at large distances as a finite expansion in Legendre
polynomials.
(b) Determine the potential explicitly at any point in space, and show
that near the origin
(c) If there exists at the origin a nucleus with a quadrupole moment
q = io~ 24 cm 2 , determine the magnitude of the interaction energy, assuming
that the unit of charge in P (r) above is the electronic charge and the unit of
length is the hydrogen Bohr radius a = h 2 lme* = 0.529 x lO" 8 cm.
Express your answer as a frequency by dividing by Planck's constant h.
The charge density in this problem is that for the m = ±1 states of the
2/7 level in hydrogen, while the quadrupole interaction is of the same order
as found in molecules.
4.4 A very long, right circular, cylindrical shell of dielectric constant e and inner
and outer radii a and b, respectively, is placed in a previously uniform
electric field E with its axis perpendicular to the field. The medium inside
and outside the cylinder has a dielectric constant of unity.
(a) Determine the potential and electric field in the three regions,
neglecting end effects.
(6) Sketch the lines of force for a typical case of b ~ 2a.
(c) Discuss the limiting forms of your solution appropriate for a solid
dielectric cylinder in a uniform field, and a cylindrical cavity in a uniform
dielectric.
4.5 A point charge q is located in free space a distance d from the center of a
dielectric sphere of radius a {a < d) and dielectric constant e.
(a) Find the potential at all points in space as an expansion in spherical
harmonics.
(Jb) Calculate the rectangular components of the electric field near the
center of the sphere.
(c) Verify that, in the limit e ► oo, your result is the same as that for the
conducting sphere.
130 Classical Electrodynamics
4.6 Two concentric conducting spheres of inner and outer radii a and b,
respectively, carry charges ± Q. The empty space between the spheres is
halffilled by a hemispherical shell of dielectric (of dielectric constant e), as
shown in the figure.
(a) Find the electric field everywhere between the spheres.
(b) Calculate the surfacecharge distribution on the inner sphere.
(c) Calculate the polarizationcharge density induced on the surface of the
dielectric at r = a.
4.7 The following data on the variation of dielectric constant with pressure are
taken from the Smithsonian Physical Tables, 9th ed., p. 424:
Air at 292°K
Pressure (atm)
20 1.0108 Relative density of
40 1.0218 air as a function of
60 1.0333
pressure is given in
80 1.0439 AIP Handbook, p.
100 1.0548 483.
Pentane (C 5 H 12 ) at 303 °K
Pressure (atm) Density (gm/cm 3 ) e
1
0.613
1.82
10 3
0.701
1.96
x 10 3
0.796
2.12
x 10 3
0.865
2.24
x 10 3
0.907
2.33
12
Test the ClausiusMossotti relation between dielectric constant and density
for air and pentane in the ranges tabulated. Does it hold exactly ? Approxi
mately? If approximately, discuss fractional variations in density and
(e — 1). For pentane, compare the ClausiusMossotti relation to the cruder
relation, (e — 1) <x density.
4.8 Water vapor is a polar gas whose dielectric constant exhibits an appreciable
temperature dependence. The following table gives experimental data on
this effect. Assuming that water vapor obeys the ideal gas law, calculate the
molecular polarizability as a function of inverse temperature and plot it.
From the slope of the curve, deduce a value for the permanent dipole
[Probs. 4] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 131
moment of the H 2 molecule (express the dipole moment in esu— stat
coulombcentimeters) .
(°K)
Pressure (cm Hg)
(e  1) x 10 5
393
56.49
400.2
423
60.93
371.7
453
65.34
348.8
483
69.75
328.7
4.9 Two long, coaxial, cylindrical conducting surfaces of radii a and b are
lowered vertically into a liquid dielectric. If the liquid rises a distance h
between the electrodes when a potential difference V is established between
them, show that the susceptibility of the liquid is
X e 
(b 2  a 2 ) P gh In (bid)
V 2
where p is the density of the liquid, g is the acceleration due to gravity, and
the susceptibility of air is neglected.
5
Magnetostatics
5.1 Introduction and Definitions
In the preceding chapters various aspects of electrostatics (i.e., the
fields and interactions of stationary charges and boundaries) have been
studied. We now turn to steadystate magnetic phenomena. From an
historical point of view, magnetic phenomena have been known and
studied for at least as long as electric phenomena. Lodestones were known
in ancient times ; the mariner's compass is a very old invention ; Gilbert's
researches on the earth as a giant magnet date from before 1600. In
contrast to electrostatics, the basic laws of magnetic fields did not follow
straightforwardly from man's earliest contact with magnetic materials.
The reasons are several, but they all stem from the radical difference
between magnetostatics and electrostatics : there are no free magnetic
charges. This means that magnetic phenomena are quite different from
electric phenomena and that for a long time no connection was established
between them. The basic entity in magnetic studies was what we now know
as a magnetic dipole. In the presence of magnetic materials the dipole
tends to align itself in a certain direction. That direction is by definition
the direction of the magneticflux density, denoted by B, provided the
dipole is sufficiently small and weak that it does not perturb the existing
field. The magnitude of the flux density can be defined by the mechanical
torque N exerted on the magnetic dipole :
N = jl x B (5.1)
where {a is the magnetic moment of the dipole, defined in some suitable
set of units. *
* In analogy with the 100 strokes of cat's fur on an amber rod, we might define our unit
of dipole strength as that of a iinch finishing nail which has been stroked slowly 100
times with a certain "standard" lodestone held in a certain standard orientation. With
a little thought we might even think of a more reliable and reproducible standard!
132
[Sect. 5.2] Magnetostatics 133
Already, in the definition of the magneticflux density B (sometimes
called the magnetic induction), we have a more complicated situation than
for the electric field. Further quantitative elucidation of magnetic
phenomena did not occur until the connection between currents and
magnetic fields was established. A current corresponds to charges in
motion and is described by a current density J, measured in units of positive
charge crossing unit area per unit time, the direction of motion of the
charges defining the direction of J. In electrostatic units, current density
is measured in statcoulombs per square centimetersecond, and is some
times called statamperes per square centimeter, while in mks units it is
measured in coulombs per square metersecond or amperes per square
meter. If the current density is confined to wires of small cross section,
we usually integrate over the crosssectional area and speak of a current
of so many statamperes or amperes flowing along the wire.
Conservation of charge demands that the charge density at any point
in space be related to the current density in that neighborhood by a
continuity equation:
d P + V • J = (5.2)
dt
This expresses the physical fact that a decrease in charge inside a small
volume with time must correspond to a flow of charge out through the
surface of the small volume, since the total number of charges must be
conserved. Steadystate magnetic phenomena are characterized by no
change in the net charge density anywhere in space. Consequently in
magnetostatics
V • J = (5.3)
We now proceed to discuss the experimental connection between current
and magneticflux density and to establish the basic laws of magneto
statics.
5.2 Biot and Savart Law
In 1819 Oersted observed that wires carrying electric currents produced
deflections of permanent magnetic dipoles placed in their neighborhood.
Thus the currents were sources of magneticflux density. Biot and Savart
(1820), first, and Ampere (18201825), in much more elaborate and
thorough experiments, established the basic experimental laws relating the
magnetic induction B to the currents and established the law of force
between one current and another. Although not in the form in which
134
Classical Electrodynamics
Fig. 5.1 Elemental magnetic induction
dB due to current element I d\.
Ampere deduced it, the basic relation is the following. If d\ is an element
of length (pointing in the direction of current flow) of a filamentary wire
which carries a current / and x is the coordinate vector from the element
of length to an observation point P, as shown in Fig. 5.1, then the
elemental flux density dB at the point P is given in magnitude and direction
by
(d\ x x)
dB = kl
x
(5.4)
It should be noted that (5.4) is an inverse square law, just as is Coulomb's
law of electrostatics. However, the vector character is very different.
If, instead of a current flowing there is a single charge q moving with a
velocity v, then the flux density will be*
B = kq — = kv x E
(5.5)
where E is the electrostatic field of the charge q. (This flux density is,
however, time varying. We shall restrict the discussions in the present
chapter to steadystate current flow.)
In (5.4) and (5.5) the constant k depends on the system of units used, as
discussed in detail in the Appendix. If current is measured in esu, but the
flux density is measured in emu, the constant is k = 1/c, where c is found
experimentally to be equal to the velocity of light in vacuo (c = 2.998 x
10 10 cm/sec). This system of units is called the Gaussian system. To insert
the velocity of light into our equations at this stage seems a little artificial,
but it has the advantage of measuring charge and current in a consistent
set of units so that the continuity equation (5.2) retains its simple form,
without factors of c. We will adopt the Gaussian system here.
Assuming that linear superposition holds, the basic law (5.4) can be
integrated to determine the magneticflux density due to various config
urations of currentcarrying wires. For example, the magnetic induction
* True only for particles moving with velocities small compared to that of light.
[Sect. 5.2]
Magnetostatics
135
B of the long straight wire shown in Fig. 5.2 carrying a current / can be
seen to be directed along the normal to the plane containing the wire and
the observation point, so that the lines of magnetic induction are concentric
circles around the wire. The magnitude of B is given by
IBI
^
dl
oo (r 2 + n
2\ 3 A
2/
cR
(5.6)
where R is the distance from the observation point to the wire. This is the
experimental result first found by Biot and Savart and is known as the
BiotSavart law. Note that the magnitude of the induction B varies with
R in the same way as the electric field due to a long line charge of uniform
linearcharge density. This analogy shows that in some circumstances
there may be a correspondence between electrostatic and magnetostatic
problems, even though the vector character of the fields is different. We
shall see more of that in later sections.
Ampere's experiments did not deal directly with the determination of
the relation between currents and magnetic induction, but were concerned
rather with the force which one currentcarrying wire experiences in the
presence of another. Since we have already introduced the idea that a
current element produces a magnetic induction, we phrase the force law as
the force experienced by a current element I x d\ x in the presence of a
magnetic induction B. The elemental force is
d¥ = 1 (dl x x B)
c
(5.7)
I x is the current in the element (measured in esu), B is the flux density (in
emu), and c is the velocity of light. If the external field B is due to a closed
current loop #2 with current 7 2 , then the total force which a closed current
Fig. 5.2
136 Classical Electrodynamics
Fig. 5.3 Two Amperian current loops,
loop #1 with current I x experiences is [from (5.4) and (5.7)]:
F M.4^i x ^ x ^ (5.8)
c 2 JJ x 12  3
The line integrals are taken around the two loops; x 12 is the vector
distance from line element d\ 2 to dl lt as shown in Fig. 5.3. This is the
mathematical statement of Ampere's observations about forces between
currentcarrying loops. By manipulating the integrand it can be put in a
form which is symmetric in dl x and d\ 2 and which explicitly satisfies
Newton's third law. Thus
d\ x x (dl 2 x x 12 ) = _ {dli . dh) _x^ + dl J d ±lM, (5 .9)
ix 12 r x 12 i \ x 12 i /
The second term involves a perfect differential in the integral over dl v
Consequently it gives no contribution to the integral (5.8), provided the
paths are closed or extend to infinity. Then Ampere's law of force between
current loops becomes
_ hh Li (<fll • ^2)Xi2
c 2 J J x 12  3
showing symmetry in the integration, apart from the necessary vectorial
dependence on x 12 .
Each of two long, parallel, straight wires a distance d apart, carrying
currents I x and I 2 , experiences a force per unit length directed perpen
dicularly towards the other wire and of magnitude,
F = ^ (5.11)
c 2 d
The force is attractive (repulsive) if the currents flow in the same (opposite)
directions. The forces which exist between currentcarrying wires can be
[Sect. 5.3] Magnetostatics 137
used to define magneticflux density in a way that is independent of per
manent magnetic dipoles.* We will see later that the torque expression
(5.1) and the force result (5.7) are intimately related.
If a current density J(x) is in an external magneticflux density B(x), the
elementary force law implies that the total force on the current distribution
is
if'
J(x) x B(x) d 3 x (5.12)
cJ
Similarly the total torque is
'x x (J x B)d 3 x (5.13)
HJ'
These general results will be applied to localized current distributions in
Section 5.6.
5.3 The Differential Equations of Magnetostatics
and Ampere's Law
The basic law (5.4) for the magnetic induction can be written down in
general form for a current density J(x) :
B(x) =  f J(x') x (x ~ x '> d 3 x> (5.14)
cJ x — x' 3
This expression for B(x) is the magnetic analog of electric field in terms of
the charge density:
E(x) = f P (x) [ X ~ X ^ V (5.15)
J x — X \ d
Just as this result for E was not as convenient in some situations as
differential equations, so (5.14) is not the most useful form for magneto
statics, even though it contains in principle a description of all the
phenomena.
In order to obtain the differential equations equivalent to (5.14) we
transform (5.14) into the form:
B(x) =  V x f J(x>) d 3 x' (5.16)
c J x _ x'
* In fact, (5.1 1) is the basis of the internationally accepted standard of current (actually
I/c here). See the Appendix.
138 Classical Electrodynamics
From (5.16) it follows immediately that the divergence of B vanishes:
VB = (5.17)
This is the first equation of magnetostatics and corresponds toVxE =
in electrostatics. By analogy with electrostatics we now calculate the curl
ofB:
VxB = VxVx f J(x>) dV (5.18)
c J x — x'l
With the identity V x (V x A) = V(V • A)  V 2 A for an arbitrary
vector field A, expression (5.18) can be transformed into
V x B = I V fj(x') • v(— i— ) dV  i f J(x')V 2 (^—) d 3 x'
c J \x — x'/ c J \x — x/
(5.19)
Using the fact that
v(_L_) _ W?)
\xx'/ \xx'/
and
V 2 ( ) = 47t6(x  x')
\x — x'/
the integrals in (5.19) can be written:
V x B =   V f J(x') • V ( ) d z x' + — J(x) (5.20)
c J \x — x'/ c
Integration by parts yields
v X B = ±r J + ivf v ' J(x ' ) dV (5.20
c c J \x — x'l
But for steadystate magnetic phenomena V • J = 0, so that we obtain
VxB = — J (5.22)
c
This is the second equation of magnetostatics, corresponding to V • E =
Airp in electrostatics.
In electrostatics Gauss's law (1.11) is the integral form of the equation
V • E = Airp. The integral equivalent of (5.22) is called Ampere's law. It
is obtained by applying Stokes's theorem to the integral of the normal
[Sect. 5.4]
Magnetostatics
139
Fig. 5.4
component of (5.22) over an open surface S bounded by a closed curve C,
as shown in Fig. 5.4. Thus
is transformed into
I V x B • n </a = — f J • n
Js c Js
<t B • d\ = — J • n da
Jc c Js
da
(5.23)
(5.24)
Since the surface integral of the current density is the total current /passing
through the closed curve C, Ampere's law can be written in the form :
B.<fl = ^I
(5.25)
Just as Gauss's law can be used for calculation of the electric field in highly
symmetric situations, so Ampere's law can be employed in analogous
circumstances.
5.4 Vector Potential
The basic differential laws of magnetostatics are given by
V x
B =
4tt
— J
c
V
B =
=
(5.26)
The problem is how to solve them. If the current density is zero in the
region of interest, V x B = permits the expression of the vector
magnetic induction B as the gradient of a magnetic scalar potential,
B = — V<I> M . Then (5.26) reduces to Laplace's equation for <b M , and all
our techniques for handling electrostatic problems can be brought to
bear. There are a large number of problems which fall into this class, but
we will defer discussion of them until later in the chapter. The reason
140 Classical Electrodynamics
is that the boundary conditions are different from those encountered in
electrostatics, and the problems usually involve macroscopic media
with magnetic properties different from free space with charges and cur
rents.
A general method of attack is to exploit the second equation in (5.26).
If V • B = everywhere, B must be the curl of some vector field A(x),
called the vector potential,
B(x) = V x A(x) (5.27)
We have, in fact, already written B in this form (5.16). Evidently, from
(5.16), the general form of A is
A(x) =  f J(x/) d z x' + VT( X ) (5.28)
c J x — x'
The added gradient of an arbitrary scalar function *F shows that, for a
given magnetic induction B, the vector potential can be freely transformed
according to
A>A+VT (5.29)
This transformation is called a gauge transformation. Such transformations
on A are possible because (5.27) specifies only the curl of A. For a
complete specification of a vector field it is necessary to state both its curl
and its divergence. The freedom of gauge transformations allows us to
make V • A have any convenient functional form we wish.
If (5.27) is substituted into the first equation in (5.26), we find
or
V x (V x A) = — J
c
V(V • A)  V 2 A = — J
c
(5.30)
If we now exploit the freedom implied by (5.29), we can make the con
venient choice of gauge, * V • A = 0. Then each rectangular component
of the vector potential satisfies Poisson's equation,
V 2 A=— J (5.31)
c
* The choice is called the Coulomb gauge, for a reason which will become apparent
only in Section 6.5.
[Sect. 5.5] Magnetostatics 141
From our discussions of electrostatics it is clear that the solution for A in
unbounded space is (5.28) with T = :
A( HI^V V (5  32)
The condition W = can be understood as follows. Our choice of gauge,
V • A = 0, reduces to V 2t F = 0, since the first term in (5.28) has zero
divergence because of V • J = 0. If V 2 T = holds in all space, Y must
vanish identically.
5.5 Vector Potential and Magnetic Induction for
a Circular Current Loop
As an illustration of the calculation of magnetic fields from given
current distributions we consider the problem of a circular loop of radius
a, lying in the xy plane, centered at the origin, and carrying a current /, as
shown in Fig. 5.5. The current density J has only a component in the <f>
direction,
J^ = /<5(cos 0') d (r ' ~ a) (5.33)
a
The delta functions restrict current flow to a ring of radius a. Only a <f>
component of J means that A will have only a (f> component also. But this
component A^ cannot be calculated by merely substituting J+ into (5.32).
Equation (5.32) holds only for rectangular components of A.* Thus we
write rectangular components of J:
J* = — J 4> sin 4>' 1
(5.34)
jy — j f COS <f> J
Since the geometry is cylindrically symmetric, we may choose the obser
vation point in the xz plane (<f> = 0) for purposes of calculation. Then it is
clear that the x component of the vector potential vanishes, leaving only
* The reason is that the vector Poisson's equation (5.31) can be treated as three
uncoupled scalar equations, V 2 A ( = (—knjc)Ji, only if the components A { , J f are
rectangular components. If A is resolved into orthogonal components with unit vectors
which are functions of position, the differential operation involved in (5.31) mixes the
components together, giving coupled equations. See Morse and Feshbach, pp. 51 and
116117.
142
Classical Electrodynamics
z
Fig. 5.5
the y component, which is A+. Thus
M r. 9) = i > dr' iff co, ^ «co. 8) *r'  ,) (5 .35)
* ca J x — x 
where x  x' = [r 2 + r' 2  2rr'(cos cos 0' + sin sin (9' cos <£')]^
We first consider the straightforward evaluation of (5.35). Integration
over the delta functions leaves the result
^(r,0) = P
c Jo (a
cos <f>' d<f>'
+ r 2 — 2ar sin cos <£')'
(5.36)
This integral can be expressed in terms of the complete elliptic integrals K
and E:
l~ (2  k 2 )K(k)  lEjkj
;L /c 2
A^r, 0) =
41a
cVa 2 + r 2 + 2arsin0'
where the argument of the elliptic integrals is
Aar sin
(5.37)
k 2 =
a 2 + r 2 + 2ar sin
The components of magnetic induction,
1 d 1
B r = — r7 ^ ( sin dA +)
rs.va.Qdd
*• — a^
#. =
(5.38)
[Sect. 5.5]
Magnetostatics
143
can also be expressed in terms of elliptic integrals. But the results are not
particularly illuminating (useful, however, for computation).
For small k 2 , corresponding to a > r, a < r, or < 1, the square
bracket in (5.37) reduces to (ttA: 2 /16). Then the vector potential becomes
approximately
AJr, 0) = — . / Sin0 • ^ (539)
c (a 2 + r 2 + 2ar sin 0)'
The corresponding fields are
B.
lira 2 Ma 2 + 2r 2 + ar sin 0)
cos
(a 2 + r 2 + 2ar sin 0)
?A
ha 2 . a ( 2a 2 r 2 + ar sin 0)
B ~ sin —r : — rrr:
e c (a 2 + r 2 + 2arsmd) A
(5.40)
These can easily be specialized to the three regions, near the axis (0 < 1),
near the center of the loop (r < a), and far from the loop (r > a).
Of particular interest are the fields far from the loop:
cos
sin
(5.41)
Comparison with the electrostatic dipole fields (4.12) shows that the
magnetic fields far away from a circular current loop are dipole in character.
By analogy with electrostatics we define the magnetic dipole moment of the
loop to be
7r/a2 (5.42)
m =
We will see in the next section that this is a special case of a general
result— localized current distributions give dipole fields at large distances;
the magnetic moment of a plane current loop is the product of the area of
the loop times 7/c.
Although we have obtained a complete solution to the problem in
terms of elliptic integrals, we will illustrate the use of a spherical harmonic
expansion to point out similarities and differences between the magneto
static and electrostatic problems. Thus we return to (5.35) and substitute
144 Classical Electrodynamics
the spherical expansion (3.70) for x — x' 1 :
4rrl„ ^Y lm (6,0)
^ = ^ReV
V
ca
^,21+1
l,m
X I r' 2 dr' dQ.' <5(cos 6') d(r'  a)e**' ^ Y* m (6', f ) (5.43)
J rt 1
The presence of e l4> ' means that only m = + 1 will contribute to the sum.
Hence
A.J. —
Sir 2 Ia
*^£M'?*M w
where now r< (r^ is the smaller (larger) of a and r. The squarebracketed
quantity is a number depending on /:
2/ + 1
4tt/(/ + 1)
JV<o) =
o,
Then ^ can be written
for J even
' (ir +1 r(n + f) l (5>45)
4tt/(/ + i)L r(n + i)r(f) J'
for / = In + 1
2/+ 1
A + c Z 2 (n + 1)! r 2 >" +2 F2 * +l(C ° S G) (546)
n =
where (2«  1)!! = (2n  1)(2«  3)( • •) x 5 X 3 x 1, and the n =
coefficient in the sum is unity by definition. To evaluate the radial com
ponent of B from (5.38) we need
f(Vl* 2 P?(*)) = /(/+ 1)^0*0
dx
(5.47)
Then we find
_ iTTla ^ (l)"(2n + l)H r 2 < n+1 ( m ( .
cr *—<
n =
2 n n\
The 6 component of B is similarly
B a =
7r/a a ^(l) n (2n + 1)!!
* 2 ^ (D w (
2L, 2\n
+ 1)!
2n +
2n +
7 3 W
w
Pj B+1 (cos 0)
(5.49)
[Sect. 5.6] Magnetostatics 145
The upper line holds for r < a, and the lower line for r > a. For r > a,
only the n = term in the series is important. Then, since P^cos 0) =
sin 0, (5.48) and (5.49) reduce to (5.41). For r < a, the leading term is
again n = 0. The fields are then equivalent to a magnetic induction
lirl/ac in the z direction, a result that can be found by elementary means.
We note a characteristic difference between this problem and a cor
responding cylindrically symmetric electrostatic problem. Associated
Legendre polynomials appear, as well as ordinary Legendre polynomials.
This can be traced to the vector character of the current and vector
potential, as opposed to the scalar properties of charge and electrostatic
potential.
Another mode of attack on the problem of the loop is to employ an
expansion in cylindrical waves. Instead of (3.70) as a representation of
x — x' _1 we may use the cylindrical form (3.148) or (3.149). The appli
cation of this technique to the circular loop will be left to the problems. It
is generally useful for any current distribution which involves current
flowing only in the <f> direction.
5.6 Magnetic Fields of a Localized Current Distribution;
Magnetic Moment
We now consider the properties of a general current distribution which
is localized in a small region of space, "small" being relative to the scale
of length of interest to the observer. The proper treatment of this problem,
in analogy with the electrostatic multipole expansion, demands a discussion
of vector spherical harmonics. These are presented in Chapter 16 in
connection with multipole radiation. We will be content here with only
the lowest order of approximation. Starting with (5.32), we expand the
denominator in powers of x' measured relative to a suitable origin in the
localized current distribution, shown schematically in Fig. 5.6:
1 = — + ^^ + • • • (5.50)
x  x' x jx 3
Then a given component of the vector potential will have the expansion,
A&) = J f J,(x') dV + * • L(x')x'dV + • • • (5.51)
cxj qxr j
For a localized steadystate current distribution the volume integral of J
vanishes because V • J = 0. Consequently the first term, corresponding to
the monopole term in an electrostatic expansion, vanishes.
146
Classical Electrodynamics
The integrand of the second term can be manipulated into a more
convenient form by using the triple vector product. Thus
(x • x') J = (x • J)x'  x x (x' x J)
(5.52)
The volume integral of the first term on the right can be shown to be the
negative of the integral of the lefthand side of (5.52). Thus we consider
the integral,
y jX ! d 3 x' = J V • (*/JK' d z x' =  f x/(J . V>/ d 3 x'
= jz/Jid 3 *' (5.53)
The step from the first integral to the second depends on V • J = 0; the
following step involves an integration by parts. With this identity (5.52)
can be written in integrated form as
J (x • x')J(x') d 3 x' = x x J [x' x J(x')] d 3 x' (5.54)
We now define the magnetic moment of the current distribution J as
= ^Jx' x J(x')d 3 x' (5.55)
m
Note that it is sometimes useful to consider the integrand in (5.55) as a
magneticmoment density or magnetization. We denote the magnetization
due to the current density J by
Jl = ^ (x x J) (5.56)
2c
The vector potential (5.51) can be expressed in terms of m as
... m x x , e e ~
AW = ^ (5.57)
This is the lowest nonvanishing term in the expansion of A for a localized
steadystate current distribution. The magnetic induction B can be
Fig. 5.6 Localized current density
J(x') gives rise to a magnetic induc
tion at the point P with coordi
nate x.
[Sect. 5.6]
Magnetostatics
141
Fig. 5.7
calculated directly by evaluating the curl of (5.57):
3„(nm)n, + ^ m3(x)
x r 3
(5.58)
Here n is a unit vector in the direction x. Since (5.57) and (5.58) have
meaning only outside the current distribution, we drop the delta function
term. The magnetic induction (5.58) has exactly the form (4.13) of the
field of a dipole. This is the generalization of the result found for the
circular loop in the last section. Far away from any localized current
distribution the magnetic induction is that of a magnetic dipole of dipole
moment given by (5.55).
If the current is confined to a plane, but otherwise arbitrary, loop, the
magnetic moment can be expressed in a simple form. If the current / flows
in a closed circuit whose line element is d\, (5.55) becomes
m = —
2c
x d\
(5.59)
For a plane loop such as that in Fig. 5.7, the magnetic moment is perpendi
cular to the plane of the loop. Since \{x x d\) = da, where da is the
triangular element of the area defined by the two ends off/1 and the origin,
the loop integral in (5.59) gives the total area of the loop. Hence the
magnetic moment has magnitude,
(5.60)
m =  X (Area)
c
regardless of the shape of the circuit.
If the current distribution is provided by a number of charged particles
with charges q t and masses M* in motion with velocities v*, the magnetic
moment can be expressed in terms of the orbital angular momentum of
the particles. The current density is
J=24iM(x Xi ) < 5  61)
148 Classical Electrodynamics
where x,. is the position of the ith particle. Then the magnetic moment
(5.55) becomes
i
The vector product (x t x v^) is proportional to the rth particle's orbital
angular momentum, L< = M^ x v,). Thus (5.62) becomes
m =2^ L * ( 5  63 )
^ *
If all the particles in motion have the same charge to mass ratio (qjM t =
e(M), the magnetic moment can be written in terms of the total orbital
angular momentum L:
m
= ^Tl,=
2Mc^ 2Mc
(5.64)
This is the wellknown classical connection between angular momentum
and magnetic moment which holds for orbital motion even on the atomic
scale. But this classical connection fails for the intrinsic moment of
electrons and other elementary particles. For electrons, the intrinsic
moment is slightly more than twice as large as implied by (5.64), with the
spin angular momentum S replacing L. Thus we speak of the electron
having a g factor of 2(1.001 17). The departure of the magnetic moment
from its classical value has its origins in relativistic and quantummechanical
effects which we cannot consider here.
5.7 Force and Torque on a Localized Current Distribution in an External
Magnetic Induction
If a localized distribution of current is placed in an external magnetic
induction B(x), it experiences forces and torques according to Ampere's
laws. The general expressions for the total force and torque are given by
(5.12) and (5.13). If the external magnetic induction varies slowly over
the region of current, a Taylor's series expansion can be utilized to find
the dominant terms in the force and torque. A component of B can be
expanded around a suitable origin,
B i (x) = B i (0) + x.VB i (0) + ... (5.65)
The force (5.12) then becomes
F =   B(0) x J J(x') dV + if J(x') x [(x' • V)B(O)] d z x' + • • • (5.66)
[Sect. 5.7] Magnetostatics 149
Since the volume integral of J vanishes for steadystate currents, the
lowestorder term is the one involving the gradient of B. Because the
integrand involves J and x, in addition to VB, we expect that the integral
can be somehow transformed into the magnetic moment (5.55). To
accomplish this we use
J x [(x' • V)B] = J x V(x' • B) =  V x [J(x' • B)] (5.67)
The first step depends on the fact that V x B = for the external field,
and that the gradient operator operates only on B. Then the force can be
written
F=Vx f J(x' • B) d 3 x' + • • • (5.68)
Use can now be made of identity (5.54) with the fixed vector x replaced by
B. Then we obtain
F = V x (B x m) = (m • V)B = V(m • B) (5.69)
where m is the magnetic moment (5.55). The second form in (5.69) follows
from V • B = 0, while the third depends on V x B = 0.
A localized current distribution in a nonuniform magnetic induction
experiences a force proportional to its magnetic moment m and given by
(5.69). One simple application of this result is the timeaverage force on a
charged particle spiraling in a nonuniform magnetic field. As is well
known, a charged particle in a uniform magnetic induction moves in a
circle at right angles to the field and with constant velocity parallel to the
field, tracing out a helical path. The circular motion is, on the time average,
equivalent to a circular loop of current which will have a magnetic moment
given by (5.60). If the field is not uniform but has a small gradient (so that
in one turn around the helix the particle does not feel significantly different
field strengths), then the motion of the particle can be discussed in terms
of the force on the equivalent magnetic moment. Consideration of the
signs of the moment and the force shows that charged particles tend to be
repelled by regions of high flux density, independent of the sign of their
charge. This is the basis of the socalled "magnetic mirrors" discussed in
Section 12.10 from another point of view.
The total torque on the localized current distribution is found in a
similar way by inserting expansion (5.65) into (5.13). Here the zeroth
order term in the expansion contributes. Keeping only this leading term,
we have
N
= if*' x [J x B(0)] d*x' (5.70)
150 Classical Electrodynamics
Writing out the triple vector product, we get
N = i J*[(x' • B)J  (x' • J)B] dV (5.71)
The first integral is the same one considered in (5.68). Hence we can write
down its value immediately. The second integral vanishes for a localized
steadystate current distribution, as can be seen from the identity,
V • (x 2 J) = 2(x • J) + x 2 V • J. The leading term in the torque is therefore
N = m x B(0) (5.72)
This is the familiar expression for the torque on a dipole, discussed in
Section 5.1 as one of the ways of defining the magnitude and direction of
the magnetic induction.
The potential energy of a permanent magnetic moment (or dipole) in
an external magnetic field can be obtained from either the force (5.69) or
the torque (5.72). If we interpret the force as the negative gradient of a
potential energy U, we find
U = m • B (5.73)
For a magnetic moment in a uniform field the torque (5.72) can be inter
preted as the negative derivative of U with respect to the angle between B
and m. This wellknown result for the potential energy of a dipole shows
that the dipole tends to orient itself parallel to the field in the position of
lowest potential energy.
We remark in passing that (5.73) is not the total energy of the magnetic
moment in the external field. In bringing the dipole m into its final
position in the field, work must be done to keep the current J which
produces m constant. Even though the final situation is a steadystate,
there is a transient period initially in which the relevant fields are time
dependent. This lies outside our present considerations. Consequently
we will leave the discussion of the energy of magnetic fields to Section 6.2,
after having treated Faraday's law of induction.
5.8 Macroscopic Equations
So far we have dealt with the basic laws (5.17) and (5.22) of steadystate
magnetic fields as microscopic equations in the sense of Chapter 4. We
have assumed that the current density J was a completely known function
of position. In macroscopic problems this is often not true. The atoms in
matter have electrons which give rise to effective atomic currents the
current density of which is a rapidly fluctuating quantity. Only its average
[Sect. 5.8] Magnetostatics 151
over a macroscopic volume is known or pertinent. Furthermore, the
atomic electrons possess intrinsic magnetic moments which cannot be
expressed in terms of a current density. These moments can give rise to
dipole fields which vary appreciably on the atomic scale of dimensions.
To treat these atomic contributions we proceed similarly to Section 4.3.
The derivation of the macroscopic equations will only be sketched here.
A somewhat more complete discussion will be given in Section 6.10. The
reason is that for timevarying fields there is a contribution to the atomic
current from the time derivative of the polarization P. Hence all the
contributions to the current appear only in the general, timedependent
problem.
The total current density can be divided into :
(a) conductioncurrent density J, representing the actual transport of
charge ;
(b) atomiccurrent density J a , representing the circulating currents
inside atoms or molecules.
The total vector potential due to all currents is
a = i f JQO * x ' + i f J .(»'> *f (5.74)
cJ x — x' cJ x — x'l
We use a small a for the microscopic vector potential, just as we used e for
the microscopic electric field in Chapter 4. For the atomic contribution
we first consider a single molecule, and then average over molecules. The
discussion proceeds exactly as in Section 5.6 for a localized current
distribution. For a molecule with center at x, the vector potential at x is
given approximately by
a mo i(x) = m ™ix(xx,) (5 ?5)
x  x/
To take into account the intrinsic magnetic moments of the electrons, as
well as the orbital contribution, we interpret m mol as the total molecular
magnetic moment. If we now sum up over all molecules, averaging as in
Section 4.3, the macroscopic vector potential can be written
A(x) _ 1 \J£L rfV + f M <*'> * <'  *'> dV (5.76)
cJ\x x'l J x  x'l 3
where M(x) is the macroscopic magnetization (magnetic moment per unit
volume) defined by
M = AT<m mol > (5*77)
where N is the number of molecules per unit volume.
152 Classical Electrodynamics
The magnetization contribution to A in (5.76) can be rewritten in a more
useful form :
JM(x') x (x ~ *'\ d z x' = (m(x') x V d\' (5.78)
J x  x' 3 J x  x'l
Then the identity, V x (<f>M) = V0 x M + <f>V x M, can be used to
obtain
r M (x<) x &Z^) to _ f^lilM to  IV x (_M_ ) to (5.79)
J xx' 3 Jxx' J \xx'/
n x IVI
The last integral can be converted to a surface integral of , and so
x  x'
vanishes if M is assumed to be mathematically well behaved and localized
within a finite volume. Combining the first term in (5.79) with the con
ductioncurrent term in (5.76), we can write the vector potential as
A(x) = lp(x) + cVxM(x) d 3,, (580)
cJ x — x I
We see that the magnetization contributes to the vector potential as an
effective current density J M :
J M = C (V x M) (5.81)
There is one questionable step in the derivation of (5.80). That is the
use of the dipole vector potential (5.75) for all molecules, even those near
the point x. If a molecule lies within a sphere of radius a few molecular
diameters d of x, its vector potential will differ appreciably from the dipole
form (5.75), being much less singular. Thus in (5.80) the contribution
from that sphere around x is in error. To estimate its importance we note
that the magnitude of the vector potential per unit volume near x is
V x M\/R, while the volume within a distance R to (R + dR) of the
point x is AttR 2 dR. Hence the contribution to A from the immediate
neighborhood of x is in error at most by an amount of the order of
d 2 1 V x M ~ (d 2 /L) (M), where L is a macroscopic dimension measuring
the spatial variation of M. Since the whole vector potential is of the order
of (M)L, the relative error made in using the dipole approximation every
where is of the order of d 2 /L 2 . This is completely negligible unless the
macroscopic length L becomes microscopic ; then the whole development
fails.
To obtain the macroscopic equivalent of the curl equation (5.22) we
calculate B from (5.80) or, what is the same thing, write down (5.22) with
the total current (J + J M ) replacing J :
V x B = — J + 4ttV x M (5.82)
c
[Sect. 5.8] Magnetostatics 153
The V x M term can be combined with B to define a new macroscopic
field H, called the magnetic field,
H = B  4ttM (583)
Then the macroscopic equations, replacing (5.26), are
Att
V x H = — J
c
(5.84)
VB =
The introduction of H as a macroscopic field is completely analogous to
the introduction of D for the electrostatic field. The macroscopic
equations (5.84) have their electrostatic counterparts,
V  D = 4 '" > 1 (5.85)
V x E = J
We emphasize that the fundamental fields are E and B. They satisfy the
homogeneous equations in (5.84) and (5.85). The derived fields, D and H,
are introduced as a matter of convenience in order to take into account in
an average way the contributions to p and J of the atomic charges and
currents.
In analogy with dielectric media we expect that the properties of magnetic
media can be described by a small number of constants characteristic of
the material. Thus in the simplest case we would expect that B and H are
proportional :
B = ^H (5.86)
where fi is a constant characteristic of the material called impermeability.*
This simple result does hold for materials other than the ferromagnetic
substances. But for these nonmagnetic materials (x generally differs from
unity by only a few parts in 10 5 (// > 1 for paramagnetic substances,
[X < 1 for diamagnetic substances). For the ferromagnetic substances,
(5.86) must be replaced by a nonlinear functional relationship,
B = F(H) (5.87)
The phenomenon of hysteresis, shown schematically in Fig. 5.8, implies
that B is not a singlevalued function of H. In fact, the function F(H)
depends on the history of preparation of the material. The incremental
permeability of //(H) is defined as the derivative of B with respect to H,
* To be consistent with the electrostatic relation D = eE, expressing the derived
quantity D as a factor times E, we should write H = //B. But traditional usage is that
of (5.86). It makes most substances have v > 1 . Perhaps that is more comforting than
H' < 1.
154
Classical Electrodynamics
Fig. 5.8 Hysteresis loop giving B in a
ferromagnetic material as a function
ofH.
assuming that B and H are parallel. For highpermeability substances,
ju(H) can be as high as 10 6 . Most untreated ferromagnetic materials have
a linear relation (5.86) between B and H for very small fields. Typical
values of initial permeability range from 10 to 10 4 .
The complicated relationship between B and H in ferromagnetic
materials makes analysis of magnetic boundary value problems inherently
more difficult than that of similar electrostatic problems. But the very
large values of permeability sometimes allow simplifying assumptions on
the boundary conditions. We will see that explicitly in the next section.
5.9 Boundary Conditions on B and H
Before we can solve magnetic boundaryvalue problems, we must
establish the boundary conditions satisfied by B and H at the interface
between two media of different magnetic properties. If a small Gaussian
Fig. 5.9
[Sect. 5.9]
Magnetostatics
155
pillbox is oriented so that its faces are in regions 1 and 2 and parallel to
the surface boundary, S, as shown in Fig. 5.9, Gauss's theorem can be
applied to V • B = to yield
(B 2 B 1 )n =
(5.88)
where n is the unit normal to the surface directed from region 1 into region
2, and the subscripts refer to values at the surface in the two media.
If we now consider a small, narrow circuit C, as shown in Fig. 5.9, with
normal n' parallel to the interface and surface S, Stokes's theorem can be
applied to the curl equation in (5.84) to give
<£ H • d\ = — I J • n'
Jc c Js
da
(5.89)
The contributions to the line integral are the tangential values of H in
the two regions, while the surface integral is proportional to the surface
current density K (charge/length x time) in the limit of vanishing width
to the loop. Thus (5.89) becomes
or
(H 2  H x ) • (n' x n) = — n'
c
n x(H 2 H!) = — K
K
(5.90)
We express these boundary conditions in terms of the magnetic field H
and the permeability fi. For simplicity assuming no surface currents, we
have
H,.ii=(^)h 1 .ii
H 2 xn = H 1 x n
(5.91)
If (jl x > fx 2 , the normal component of H 2 is much larger than the normal
component of H l5 as shown in Fig. 5.10. In the limit (/uj^ ** °°> tne
ll^i:»M2lllllll
EH
Fig. 5.10
156 Classical Electrodynamics
magnetic field H 2 is normal to the boundary surface, independent of the
direction of H x (barring the exceptional case of H x exactly parallel to the
interface). The boundary condition on H at the surface of a very high
permeability material is thus the same as for the electric field at the surface
of a conductor. We may therefore use electrostatic potential theory for
the magnetic field. The surfaces of the highpermeability material are
approximately "equipotentials," and the lines of H are normal to these
equipotentials. This analogy is exploited in many magnetdesign problems.
The type of field is decided upon, and the pole faces are shaped to be
equipotential surfaces.
5.10 Uniformly Magnetized Sphere
To illustrate the different methods possible for the solution of a
boundary value problem in magnetostatics, we consider in Fig. 5.11 the
simple problem of a sphere of radius a, with a uniform permanent
magnetization M of magnitude M and parallel to the z axis, embedded in
a nonpermeable medium. Outside the sphere, V • B = V x B = 0.
Consequently, for r > a, B = H can be written as the negative gradient of
a magnetic scalar potential which satisfies Laplace's equation,
B ut= VO 1
(5.92)
V 2 O M = J
With the boundary condition that B — >■ for r — ► oo, the general solution
for the potential is ^
W) ^ a ,M (5 . 93)
1 =
Past experience tells us that only the lowest few terms in this expansion will
appear, probably just / = 1 .
Inside a magnetized object we cannot in general use equations (5.92)
because VxB^O. This causes no difficulty in the present simple situation
because (5.83) implies that B, H, and M are all parallel in the absence of
applied fields.
M = Mo63
Fig. 5.11
[Sect. 5.10]
Hence we assume that
Magnetostatics
B in = B o € z
Hi D = CB  477M )€3
157
(5.94)
The boundary conditions at the surface of the sphere are that B r and H d
be continuous. Thus, from (5.92), (5.93), and (5.94), we obtain
^ (/ + l)a t P,(cos 0)
B cos =2^ ^Pi
1 =
(fl  4ttM ) sin = ^ ^Ti ^
(5.95)
Evidently only the / = 1 term survives in the expansion. We find the
unknown constants ol x and B to be
4tt w o
a, = — M a J
1 3
5 = ^M
(5.96)
The fields outside the sphere are those of a dipole (5.41) of dipole moment,
(5.97)
47T o_ ,
m = — afM
3
The fields inside are
B ln = ^M
Hm= t^M
(5.98)
We note that B in is parallel to M, while H in is antiparallel. The lines of B
and H are shown in Fig. 5.12. The lines of B are continuous closed paths,
but those of H terminate on the surface. The surface appears to have a
"magneticcharge" density on it. This fictitious charge is related to the
divergence of the magnetization (see below).
The solution both inside and outside the sphere could have been
obtained from electrostatic potential theory if we had chosen to discuss H
rather than B. We can treat the equations,
V x H =
VH= 477VMJ
(5.99)
158
Classical Electrodynamics
H
Fig. 5.12 Lines of B and lines of H for a uniformly magnetized sphere. The lines of B
are closed curves, but the lines of H originate on the surface of the sphere where the
magnetic "charge," —V • M, resides.
These equations show that H is derivable from a potential, and that
— V • M acts as a magneticcharge density. Thus, with H = — V® M , we
find
V 2 ^ = 4ttV • M (5.100)
Since M is constant in magnitude and direction, its divergence is zero
inside the sphere. But there is a contribution because M vanishes outside
the sphere. We write the solution for Q> M inside and outside the sphere as
<1>m(x) =  f y /,M( ^ } dV (5.101)
J x  X 
Then we use the vector identity V • (<f>M) = M • V<f> + <£V • M to obtain
<*>m(x) =  f V • 55*1 d*x' + f M(x') • V'(— L) dV (5.102)
J x — x  J \x — X /
The first integral vanishes on integration over any volume containing the
sphere. If we convert the derivative with respect to x' into one with
respect to x according to the rule V > — V when operating on any
function of x — x', the potential can be written
0> M (x) =  V • f M(X,) d z x' = V • [m € 3 f V 2 dr' [dQ.' 1
J x  x' L ° 3 J J x  x'J
(5.103)
[Sect. 5.10] Magnetostatics 159
Only the / = part of x — x'l 1 contributes to the integral. Therefore
O M (x)=4 7 rM V.[e 3 £^] (5.104)
The integral yields different values, depending on whether r lies inside or
outside the sphere. We find easily
*«(x) = ^f^(^)cose (5.105)
where r< (r>) is the smaller (larger) of r and a. This potential yields a
dipole field outside with magnetic moment (5.97) and the constant value
H in (5.98) inside, in agreement with the first method of solution.*
Finally we solve the problem using the generally applicable vector
potential. Referring to (5.80), we see that the vector potential is given by
A(x) f^xMxO^, (5106)
J x — x'
Since M is constant inside the sphere, the curl vanishes there. But because
of the discontinuity of M at the surface, there is a surface integral contri
bution to A. If we consider (5.79), the required surface integral can be
recovered :
A(x)   f V x ( M(X,) W = i M(X>) X ° da' (5.107)
W J \x  x'/ J x  x'l
The quantity c(M x n) can be considered as a surfacecurrent density.
The equivalence of a uniform magnetization throughout a certain volume
to a surfacecurrent density c(M x n) over its surface is a general result
for arbitrarily shaped volumes. This equivalence is often useful in treating
fields due to permanent magnets.
For the sphere with M in the z direction, (M x n) has only an azimuthal
component,
(M x n)^ = M sin 0' (5.108)
To determine A we choose our observation point in the xz plane for
calculational convenience, just as in Sections 5.5. Then only the y com
ponent of — (n x M) enters. The azimuthal component of the vector
potential is then
^(x) = M a4rfQ S ; ne ' C( */' (5.109)
J x — x I
* The development from (5.101) to (5.105) is unnecessarily complicated for the simple
calculation at hand. For the uniformly magnetized sphere it is easy to show that
V M = M o cos0<5(r  a). Substitution into (5.101) and use of (3.70) yields
(5.105) directly. Equation (5.103) is still useful, of course, for more complicated
distributions of magnetization.
160 Classical Electrodynamics
where x' has coordinates (a, 0\ <f>'). The angular factor can be written
sin 0' cos <j>' = J— Re[ Y ltl (6', f )] (5.110)
Thus with expansion (3.70) for x — x' only the / = 1, m = 1 term will
survive. Consequently
A+(x) = j M a*(^) sind (5.111)
where r < (r > ) is the smaller (larger) of r and a. With only a <f> component
of A, the components of the magnetic induction B are given by (5.38).
Equation (5.111) evidently gives the uniform B inside and the dipole field
outside, as found before.
The different techniques used here illustrate the variety of ways of
solving steadystate magnetic problems, in this case with a specified
distribution of magnetization. The scalar potential method is applicable
provided no currents are present. But for the general problem with
currents we must use the vector potential (apart from special techniques
for particularly simple geometries).
5.11 Magnetized Sphere in an External Field; Permanent Magnets
In Section 5.10 we discussed the fields due to a uniformly magnetized
sphere. Because of the linearity of the field equations we can superpose a
uniform magnetic induction 60=110 throughout all space. Then we have
the problem of a uniformly magnetized sphere in an external field. From
(5.98) we find that the magnetic induction and field inside the sphere are
now
B in = B + — M
3
H in = B  jM
(5.112)
We now imagine that the sphere is not a permanently magnetized object,
but rather a paramagnetic or diamagnetic substance of permeability /u.
Then the magnetization M is a result of the application of the external
field. To find the magnitude of M we use (5.86):
B in = /*H in (5.113)
[Sect. 5.11]
Magnetostatics
161
(5.114)
(5.115)
Thus Fig ' 513
b + m = / ,(b ^m)
This gives a magnetization,
M = M^^)b
4ttV + 2/
We note that this is completely analogous to the polarization P of a
dielectric sphere in a uniform electric field (4.63).
For a ferromagnetic substance the arguments of the last paragraph fail.
Equation (5.115) implies that the magnetization vanishes when the
external field vanishes. The existence of permanent magnets contradicts
this result. The nonlinear relation (5.87) and the phenomenon of hysteresis
allow the creation of permanent magnets. We can solve equations (5.112)
for one relation between H in and B in by eliminating M:
B in + 2H in = 3B
(5.116)
The hysteresis curve provides the other relation between B in and H in , so
that specific values can be found for any external field. Equation (5.116)
corresponds to lines with slope —2 on the hysteresis diagram with inter
cepts 3B on the y axis, as in Fig. 5.13. Suppose, for example, that the
external field is increased until the ferromagnetic sphere becomes saturated
and decreased to zero. The internal B and H will then be given by the
point marked P in Fig. 5.13. The magnetization can be found from (5.112)
with B = 0.
The relation (5.116) between B in and H in is specific to the sphere. For
other geometries other relations pertain. The problem of the ellipsoid can
be solved exactly and shows that the slope of the lines (5.116) range from
zero for a flat disc to — oo for a long needlelike object. Thus a larger
internal magnetic induction can be obtained with a rod geometry than
with spherical or oblate spheroidal shapes.
1<>2 Classical Electrodynamics
5.12 Magnetic Shielding; Spherical Shell of Permeable Material in a
Uniform Field
Suppose that a certain magnetic induction B exists in a region of empty
space initially. A permeable body is now placed in the region. The lines
of magnetic induction are modified. From our remarks at the end of
Section 5.9 concerning media of very high permeability we would expect
that the field lines would tend to be normal to the surface of the body.
Carrying the analogy with conductors further, if the body is hollow, we
would expect that the field in the cavity would be smaller than the external
field, vanishing in the limit fj, ► oo. Such a reduction in field is said to be
due to the magnetic shielding provided by the permeable material. It is of
considerable practical importance, since essentially fieldfree regions are
often necessary or desirable for experimental purposes or for the reliable
working of electronic devices.
As an example of the phenomenon of magnetic shielding we consider a
spherical shell of inner (outer) radius a (b), made of material of perme
ability p, and placed in a formerly uniform constant magnetic induction
B , as shown in Fig. 5.14. We wish to find the fields B and H everywhere
in space, but most particularly in the cavity (r < a), as functions of p.
Since there are no currents present, the magnetic field H is derivable from
a scalar potential, H = VO M . Furthermore, since B = ^H, the
divergence equation V . B = becomes V . H = in the various regions.
Thus the potential <b M satisfies Laplace's equation everywhere. The
problem reduces to finding the proper solutions in the different regions to
satisfy the boundary conditions (5.88) and (5.90) at r = a and r = b.
For r>b, the potential must be of the form,
oo
®m = ~B r cos + J ik ^(cos 0) (5.117)
Fig. 5.14
[Sect. 5.12]
Magnetostatics
163
in order to give the uniform field, H = B = B , at large distances. For
the inner regions, the potential must be
a < r <
r < a
z=o r
®M = I*ir l P l (cos6)
1 =
(5.118)
The boundary conditions at r — a and r = b are that H e and B r be
continuous. In terms of the potential O m these conditions become
9 *( M =^( M
do
dd
*>* (fl+) _??*(«_)
do
dd
or or
V
*>"<«,) = ?*«(«_)
dr
dr
(5.119)
The notation b ± means the limit r— ►& approached from r ^ b, and
similarly for a ± . These four conditions, which hold for all angles 6, are
sufficient to determine the unknown constants in (5.117) and (5.118). All
coefficients with / ^ 1 vanish. The / = 1 coefficients satisfy the four
simultaneous equations
"i #7*i 7i =^o
2a t + fibPp!  2 f ,iy 1 = b^B^
/^a 3 ^! — 2/^7! — cfid 1 = 0.
The solutions for a x and 5 X are
(2/j + !)(//  1)
(5.120)
(2p + 1)0* + 2) 2^i/i l) 2
(fc 3  « 3 )5
<5x=
9^
(2 /1 + 1)( / i + 2)2"( /< 1) 2
Bn
(5.121)
The potential outside the spherical shell corresponds to a uniform field
B plus a dipole field (5.41) with dipole moment <x x oriented parallel to B .
Inside the cavity, there is a uniform magnetic field parallel to B and
equal in magnitude to — d v For /u > 1, the dipole moment a. x and the
164
Classical Electrodynamics
Fig. 5.15 Shielding effect of a shell of highly permeable material.
inner field — d ± become
a x — b%
*i
ty
7 3\ B o
(5.122)
We see that the inner field is proportional to fir 1 . Consequently a shield
made of highpermeability material with fi ~ 10 3 to 10 6 causes a great
reduction in the field inside it, even with a relatively thin shell. Figure
5.15 shows the behavior of the lines of B. The lines tend to pass through
the permeable medium if possible.
REFERENCES AND SUGGESTED READING
Problems in steadystate current flow in an extended resistive medium are analogous to
electrostatic potential problems, with the current density replacing the displacement and
the conductivity replacing the dielectric constant. But the boundary conditons are
generally different. Steadystate current flow is treated in
Jeans, Chapters IX and X,
Smythe, Chapter VI.
[Probs. 5] Magnetostatics 165
Magnetic fields due to specified current distributions and boundaryvalue problems in
magnetostatics are discussed, with numerous examples, by
Durand, Chapters XIV and XV,
Smythe, Chapters VII and XII.
The atomic theory of magnetic properties rightly falls in the domain of quantum
mechanics. Semiclassical discussions are given by
Abraham and Becker, Band II, Sections 2934,
Durand, pp. 551573, and Chapter XVII,
Landau and Lifshitz, Electrodynamics of Continuous Media,
Rosenfeld, Chapter IV.
Quantummechanical treatments appear in books devoted entirely to the electrical and
magnetic properties of matter, such as
Van Vleck.
PROBLEMS
5.1 Starting with the differential expression
Id\ X x
dB = ^~
c x A
for the magnetic induction produced by an increment / d\ of current, show
explicitly that for a closed loop carrying a current J the magnetic induction
at an observation point P is
B =   VQ
c
where O is the solid angle subtended by the loop at the point P. This is an
alternative form of Ampere's law for current loops.
5.2 (a) For a solenoid wound with N turns per unit length and carrying a
current /, show that the magneticflux density on the axis is given approxi
mately by
2nNI , „ .
B z = (cos d 1 + cos 2 )
where the angles are defined in the figure.
(a)
(b) For a long solenoid of length L and radius a show that near the axis
and near the center of the solenoid the magnetic induction is mainly parallel
to the axis, but has a small radial component
9677JV7/a 2 z/A
*'~—{if)
correct to order a 2 /L 2 , and for z < L, p < a. The coordinate z is measured
from the center point of the axis.
166 Classical Electrodynamics
(c) Show that at the end of a long solenoid the magnetic induction near
the axis has components
2nNI ttNI/p
B~ ^ , 5„ ^ —
5.3 A cylindrical conductor of radius a has a hole of radius b bored parallel to,
and centered a distance </from, the cylinder axis (d + b < a). The current
density is uniform throughout the remaining metal of the cylinder and is
parallel to the axis. Use Ampere's law and the principle of linear super
position to find the magnitude and the direction of the magneticflux
density in the hole.
5.4 A circular current loop of radius a carrying a current / lies in the xy plane
with its center at the origin.
(a) Show that the only nonvanishing component of the vector potential is
Ala f °°
A^ip, z) = dk cos kz Ijikp^Kjilcp^
where p< (p>) is the smaller (larger) of a and p.
(b) Show that an alternative expression for A+ is
A+(p, z) = — dk e*'*l J^Ukp)
(c) Write down integral expressions for the components of magnetic
induction, using the expressions of (a) and (b). Evaluate explicitly the
components of B on the axis by performing the necessary integrations.
5.5 Two concentric circular loops of radii a, b and currents /, /', respectively
{b < a), have an angle a between their planes. Show that the torque on one
of the loops is about the line of intersection of the two planes containing
the loops and has the magnitude:
mr 2^ll'b 2 ^ (» + 1)
N =
2
ac 2 Z, {In + 1)
71=0
T(n + f ) 
T(n + 2) T(f )
2n
" ^2n+l(COSa)
where iY(cos a) is an associated Legendre polynomial. Determine the
sense of the torque for a an acute angle and the currents in the same
(opposite) directions.
5.6 A sphere of radius a carries a uniform charge distribution on its surface.
The sphere is rotated about a diameter with constant angular velocity co.
Find the vector potential and magneticflux density both inside and outside
the sphere.
5.7 A long, hollow, right circular cylinder of inner (outer) radius a (b), and of
relative permeability /j,, is placed in a region of initially uniform magnetic
flux density B at right angles to the field. Find the flux density at all points
in space, and sketch the logarithm of the ratio of the magnitudes of B on the
cylinder axis to B as a function of log 10 /u for a 2 /b 2 — 0.5, 0. 1 . Neglect end
effects.
5.8 A current distribution J(x) exists in a medium of unit permeability adjacent
to a semiinfinite slab of material having permeability n and filling the
halfspace, z < 0.
[Probs. 5] Magnetostatics 167
(a) Show that for z > the magnetic induction can be calculated by
replacing the medium of permeability ^ by an image current distribution,
J*, with components,
(b) Show that for z < the magnetic induction appears to be due to a
current distribution ( (\ I J in a medium of unit permeability.
5.9 A circular loop of wire having a radius a and carrying a current / is located
in vacuum with its center a distance d away from a semiinfinite slab of
permeability /u. Find the force acting on the loop when
(a) the plane of the loop is parallel to the face of the slab,
(b) the plane of the loop is perpendicular to the face of the slab.
(c) Determine the limiting form of your answers to (a) and (b) when
d > a. Can you obtain these limiting values in some simple and direct
way?
5.10 A magnetically "hard" material is in the shape of a right circular cylinder
of length L and radius a. The cylinder has a permanent magnetization M ,
uniform throughout its volume and parallel to its axis.
(a) Determine the magnetic field H and magnetic induction B at all
points on the axis of the cylinder, both inside and outside.
O) Plot the ratios B/4ttM and H/4ttM on the axis as functions of z for
L/a = 5.
5.11 (a) Starting from the force equation (5.12) and the fact that a magnetiza
tion M is equivalent to a current density J M = c(V X M), show that, in
the absence of macroscopic currents, the total magnetic force on a body
with magnetization M can be written
/■
(V • M)B e cPx
where B e is the magnetic induction due to all other except the one in
question.
(b) Show that an alternative expression for the total force is
;■
(V • M)H d 3 x
where H is the total magnetic field, including the field of the magnetized
body.
Hint: The results of (a) and (b) differ by a selfforce term which can be
omitted (why?).
5.12 A magnetostatic field is due entirely to a localized distribution of permanent
magnetization.
(a) Show that
B • H d z x =
provided the integral is taken over all space.
!■
168 Classical Electrodynamics
ib) From the potential energy (5.73) of a dipole in an external field show
that for a continuous distribution of permanent magnetization the magneto
static energy can be written
W
= ! (h • Ud 3 x = i MM Hdh
apart from an additive constant which is independent of the orientation or
position of the various constituent magnetized bodies.
5.13 Show that in general a long, straight bar of uniform crosssectional area A
with uniform lengthwise magnetization M, when placed with its flat end
against an infinitely permeable flat surface, adheres with a force given
approximately by
F ~ 2nAM 2
5.14 A right circular cylinder of length L and radius a has a uniform lengthwise
magnetization M.
(a) Show that, when it is placed with its flat end against an infinitely
permeable plane surface, it adheres with a force
F = SiraLM 2
where
K(k)  E(k) K(k^)  E(kJ
K
* / . „ = > "a —
V4a 2 + L 2 ' Va 2 + L 2
(b) Find the limiting form for the force if L :> a.
6
Time Varying Fields,
Maxwell's Equations,
Conservation Laws
In the previous chapters we have dealt with steadystate problems
in electricity and in magnetism. Similar mathematical techniques were
employed, but electric and magnetic phenomena were treated as indepen
dent. The only link between them was the fact that currents which produce
magnetic fields are basically electrical in character, being charges in motion.
The almost independent nature of electric and magnetic phenomena
disappears when we consider timedependent problems. Timevarying
magnetic fields give rise to electric fields and viceversa. We then must
speak of electromagnetic fields, rather than electric or magnetic fields. The
full import of the interconnection between electric and magnetic fields
and their essential sameness becomes clear only within the framework
of special relativity (Chapter 1 1). For the present we will content ourselves
with examining the basic phenomena and deducing the set of equations
known as Maxwell's equations, which describe the behavior of electro
magnetic fields. General properties of these equations will be established
so that the basic groundwork of electrodynamics will have been laid.
Subsequent chapters will then explore the many ramifications.
In our desire to proceed to other things, we will leave out a number of
topics which, while of interest in themselves, can be studied elsewhere.
Some of these are quasistationary fields, circuit theory, inductance
calculations, eddy currents, and induction heating. None of these subjects
involves new concepts beyond what are developed in this chapter and
previous ones. The interested reader will find references at the end of the
chapter.
169
170 Classical Electrodynamics
6.1 Faraday's Law of Induction
The first quantitative observations relating timedependent electric and
magnetic fields were made by Faraday (1831) in experiments on the
behavior of currents in circuits placed in time varying magnetic fields. It
was observed by Faraday that a transient current is induced in a circuit
if (a) the steady current flowing in an adjacent circuit is turned on or off, (b)
the adjacent circuit with a steady current flowing is moved relative to the
first circuit, (c) a permanent magnet is thrust into or out of the circuit. No
current flows unless either the adjacent current changes or there is relative
motion. Faraday interpreted the transient current flow as being due to a
changing magnetic flux linked by the circuit. The changing flux induces
an electric field around the circuit, the line integral of which is called the
electromotive force, $. The electromotive force causes a current flow,
according to Ohm's law.
We now express Faraday's observations in quantitative mathematical
terms. Let the circuit C be bounded by an open surface S with unit normal
n, as in Fig. 6.1. The magnetic induction in the neighborhood of the
circuit is B. The magnetic flux linking the circuit is defined by
F = Bn da (6.1)
Js
The electromotive force around the circuit is
£=i>E'>dl (6.2)
where E' is the electric field at the element d\ of the circuit C. Faraday's
observations are summed up in the mathematical law,
dF
£=k— (6.3)
dt
The induced electromotive force around the circuit is proportional to the
time rate of change of magnetic flux linking the circuit. The sign is
specified by Lenz's law, which states that the induced current (and
accompanying magnetic flux) is in such a direction as to oppose the change
of flux through the circuit.
The constant of proportionality k depends on the choice of units for the
electric and magnetic field quantities. It is not, as might at first be
supposed, an independent empirical constant to be determined from
experiment. As we will see immediately, once the units and dimensions in
[Sect. 6.1] Time Varying Fields, Maxwell's Equations, Conservation Laws 111
Fig. 6.1
Ampere's law have been chosen, the magnitude and dimensions of A: follow
from the assumption of Galilean invariance for Faraday's law. For
Gaussian units, k = c~ x , where c is the velocity of light.
Before the development of special relativity (and even afterwards, when
dealing with relative speeds small compared with the velocity of light), it
was understood, although not often explicitly stated, by all physicists that
physical laws should be invariant under Galilean transformations. That
is, physical phenomena are the same when viewed by two observers
moving with a constant velocity v relative to one another, provided the
coordinates in space and time are related by the Galilean transformation,
x' = x +\t, t' = t. In particular, consider Faraday's observations. It is
obvious (i.e., experimentally verified) that the same current is induced in a
circuit whether it is moved while the circuit through which current is
flowing is stationary or it is held fixed while the currentcarrying circuit is
moved in the same relative manner.
Let us now consider Faraday's law for a moving circuit and see the
consequences of Galilean invariance. Expressing (6.3) in terms of the
integrals over E' and B, we have
<pE' dl= k— Bnda (6.4)
Jc dt Js
The induced electromotive force is proportional to the total time derivative
of the flux — the flux can be changed by changing the magnetic induction
or by changing the shape or orientation or position of the circuit. In form
(6.4) we have a farreaching generalization of Faraday's law. The circuit
C can be thought of as any closed geometrical path in space, not necessarily
coincident with an electric circuit. Then (6.4) becomes a relation between
the fields themselves. It is important to note, however, that the electric
field, E' is the electric field at d\ in the coordinate system in which d\ is at
rest, since it is that field which causes current to flow if a circuit is actually
present.
172 Classical Electrodynamics
If the circuit C is moving with a velocity v in some direction, as shown in
Fig. 6.2, the total time derivative in (6.4) must take into account this
motion. The flux through the circuit may change because (a) the flux
changes with time at a point, or (b) the translation of the circuit changes
the location of the boundary. It is easy to show that the result for the total
time derivative of flux through the moving circuit is*
— \Bnda = \^nda + <k(Bxv)'dl (6.5)
dt Js Js dt Jc
Equation (6.4) can now be written in the form,
(f> [E'  fe(v x B)] • d\ = k f — • n da (6.6)
Jc Js dt
This is an equivalent statement of Faraday's law applied to the moving
circuit C. But we can choose to interpret it differently. We can think of
the circuit C and surface S as instantaneously at a certain position in space
in the laboratory. Applying Faraday's law (6.4) to that fixed circuit, we
find
i>Edl= k\ —nda (6.7)
Jc Js dt
where E is now the electric field in the laboratory. The assumption of
Galilean invariance implies that the lefthand sides of (6.6) and (6.7) must
be equal. This means that the electric field E' in the moving coordinate
system of the circuit is
E' = E + k(y x B) (6.8)
To determine the constant k we merely observe the significance of E\ A
charged particle (e.g., one of the conduction electrons) in a moving circuit
Fig. 6.2
1
For a general vector field there is an added term, I (V • B)v • n da, which gives the
Js
contribution of the sources of the vector field swept over by the moving circuit. The
general result follows most easily from the use of the convective derivative,
d d
T=+VV
dt dt
[Sect. 6.2] TimeVarying Fields, Maxwell's Equations, Conservation Laws 173
will experience a force qE'. When viewed from the laboratory, the charge
represents a current J = qs d(x — x ). From the magnetic force law (5.7)
or (5.12) it is evident that this current experiences a force in agreement
with (6.8) provided the constant k is equal to c _1 .
We have thus reached the conclusion that Faraday's law takes the form
(f> E'dl =  \*>nda (6.9)
Jc c dt Js
where E' is the electric field at d\ in its rest frame of coordinates. The time
derivative on the right is a total time derivative. If the circuit C is moving
with a velocity v, the electric field in the moving frame is
E' = E +  (v x B) (6.10)
c
These considerations are valid only for nonrelativistic velocities. Galilean
invariance is not rigorously valid, but holds only for relative velocities
small compared to the velocity of light. Expression (6.10) is correct to
first order in vjc, but in error by terms of order v 2 /c 2 (see Section 11.10).
Evidently, for laboratory experiments with macroscopic circuits, (6.9) and
(6.10) are completely adequate.
Faraday's law (6.9) can be put in differential form by use of Stokes's
theorem, provided the circuit is held fixed in the chosen reference frame
(in order to have E and B defined in the same frame). The transformation
of the electromotive force integral into a surface integral leads to
/,(
VxE +  — )nda=0
s\ c dt
Since the circuit C and bounding surface S are arbitrary, the integrand
must vanish at all points in space.
Thus the differential form of Faraday's law is
VXE +  = (6.11)
c dt
We note that this is the timedependent generalization of the statement,
V x E = 0, for electrostatic fields.
6.2 Energy in the Magnetic Field
In discussing steadystate magnetic fields in Chapter 5 we avoided the
question of field energy and energy density. The reason was that the
creation of a steadystate configuration of currents and associated magnetic
174 Classical Electrodynamics
fields involves an initial transient period during which the currents and
fields are brought from zero to their final values. For such timevarying
fields there are induced electromotive forces which cause the sources of
current to do work. Since the energy in the field is by definition the total
work done to establish it, we must consider these contributions.
Suppose for a moment that we have only a single circuit with a constant
current / flowing in it. If the flux through the circuit changes, an electro
motive force $ is induced around it. In order to keep the current constant,
the sources of current must do work at the rate,
dt c dt
This is in addition to ohmic losses in the circuit which are not to be
included in the magneticenergy content. Thus, if the flux change through
a circuit carrying a current / is 6F, the work done by the sources is
6W=I6F
c
Now we consider the problem of the work done in establishing a general
steadystate distribution of currents and fields. We can imagine that the
buildup process occurs at an infinitesimal rate so that V • J = holds to
any desired degree of accuracy. Then the current distribution can be
broken up into a network of elementary current loops, the typical one of
which is an elemental tube of current of crosssectional area Ac following
a closed path C and spanned by a surface 5" with normal n, as shown in
Fig. 6.3.
We can express the increment of work done against the induced emf in
terms of the change in magnetic induction through the loop :
A(dW) = t^\ndBda
c Js
where the extra A comes from the fact that we are considering only one
elemental circuit. If we express B in terms of the vector potential A,
then we have
c Js
A(dW) = =Z (V x <5A) • n da
c Js
With application of Stokes's theorem this can be written
A(W) = J^&>SKd\
C Jn
[Sect. 6.2] Time Varying Fields, Maxwell's Equations, Conservation Laws 175
Fig. 6.3 Distribution of current
density broken up into elemental
current loops.
But J Act d\ is equal to J d 3 x, by definition, since d\ is parallel to J.
Evidently the sum over all such elemental loops will be the volume
integral. Hence the total increment of work done by the external sources
due to a change 6A(x) in the vector potential is
6W
 l .S
dA • J d 3 x
(6.12)
An expression involving the magnetic fields rather than J and dA can be
obtained by using Ampere's law :
Att
V x H= — J
Then
dW=— <5A.(V x U)d 3 x
Att J
The vector identity,
V • (P x Q) = Q • (V x P)  P • (V x Q)
can be used to transform (6.13):
(6.13)
dW
= J_ f[H . (V x <5A) + V • (H x dA)] d 3 x (6.14)
Att J
If the field distribution is assumed to be localized, the second integral
vanishes. With the definition of B in terms of A, the energy increment can
be written :
6W
= Lh
Att J
• 6B d 3 x
(6.15)
This relation is the magnetic equivalent of the electrostatic equation (4.89).
In its present form it is applicable to all magnetic media, including ferro
magnetic substances. If we assume that the medium is para or dia
magnetic, so that a linear relation exists between H and B, then
H • 6B = £ (5(H . B)
176 Classical Electrodynamics
If we now bring the fields up from zero to their final values, the total
magnetic energy will be
W=— \nBd 3 x (6.16)
This is the magnetic analog of (4.92).
The magnetic equivalent of (4.86) where the electrostatic energy is
expressed in terms of charge density and potential, can be obtained from
(6.12) by assuming a linear relation between J and A. Then we find the
magnetic energy to be
W=—]J'Ad z x (6.17)
2c J
The magnetic problem of the change in energy when an object of
permeability fi x is placed in a magnetic field whose current sources are
fixed can be treated in close analogy with the electrostatic problem of
Section 4.8. The role of E is taken by B, that of D by H. The original
medium has permeability ju and existing magnetic induction B . After
the object is in place the fields are B and H. It is left as an exercise for the
reader to verify that for fixed sources of the field the change in energy is
1^=— (BH HB )d 3 :r (6.18)
S7TJVi
where the integration is over the volume of the object. This can be written
in the alternative forms :
W= i f (^  ^H • Ho d*x = i f (1  i) B • B d*x (6.19)
Both fi x and /u can be functions of position, but they are assumed inde
pendent of field strength.
If the object is in otherwise free space (/u = 1), the change in energy can
be expressed in terms of the magnetization as
W = I M • B d 3 x (6.20)
JVx
It should be noted that (6.20) is equivalent to the electrostatic result
(4.96), except for sign. This sign change arises because the energy W
consists of the total energy change occurring when the permeable body is
introduced in the field, including the work done by the sources against the
induced electromotive forces. In this respect the magnetic problem with
fixed currents is analogous to the electrostatic problem with fixed potentials
on the surfaces which determine the fields. By an analysis equivalent to
[Sect. 6.3] TimeVarying Fields, Maxwell's Equations, Conservation Laws 177
that at the end of Section 4.8 we can show that for a small displacement
the work done against the induced emf 's is twice as large as, and of the
opposite sign to, the potentialenergy change of the body. Thus, to find
the force acting on the body, we consider a generalized displacement £ and
calculate the positive derivative of W with respect to the displacement:
'.©,
The subscript J implies fixed source currents.
The difference between (6.20) and the potential energy (5.73) for a
permanent magnetic moment in an external field (apart from the factor \,
which is traced to the linear relation assumed between M and B) comes
from the fact that (6.20) is the total energy required to produce the con
figuration, whereas (5.73) includes only the work done in establishing the
permanent magnetic moment in the field, not the work done in creating
the magnetic moment and keeping it permanent.
6.3 Maxwell's Displacement Current; Maxwell's Equations
The basic laws of electricity and magnetism which we have discussed so
far can be summarized in differential form by these four equations :
Coulomb's law: V • D = Anp
Att
Ampere's law: V x H = — J
c
 (6.22)
Faraday's law: V x E +  — =
c dt
Absence of free magnetic poles : V • B =
These equations are written in macroscopic form and in Gaussian units.
Let us recall that all but Faraday's law were derived from steadystate
observations. Consequently, from a logical point of view there is no a
priori reason to expect that the static equations hold unchanged for time
dependent fields. In fact, the equations in set (6.22) are inconsistent as
they stand.
It required the genius of J. C. Maxwell, spurred on by Faraday's
observations, to see the inconsistency in equations (6.22) and to modify
them into a consistent set which implied new physical phenomena, at that
time unknown but subsequently verified in all details by experiment. For
this brilliant stroke in 1865, the modified set of equations is justly known as
Maxwell's equations.
178 Classical Electrodynamics
The faulty equation is Ampere's law. It was derived for steadystate
current phenomena with V • J = 0. This requirement on the divergence
of J is contained right in Ampere's law, as can be seen by taking the diver
gence of both sides :
V.J = V.(VxH)sO (6.23)
While V • J = is valid for steadystate problems, the complete relation
is given by the continuity equation for charge and current :
V • J + f£ = (6.24)
ot
What Maxwell saw was that the continuity equation could be converted
into a vanishing divergence by using Coulomb's law (6.22). Thus
Then Maxwell replaced J in Ampere's law by its generalization,
J ^ J + ^¥ (6  26 >
for timedependent fields. Thus Ampere's law became
VXH = ^ J + If (6 . 27)
c c ot
still the same, experimentally verified, law for steadystate phenomena,
but now mathematically consistent with the continuity equation (6.24) for
timedependent fields. Maxwell called the added term in (6.26) the
displacement current. This necessary addition to Ampere's law is of crucial
importance for rapidly fluctuating fields. Without it there would be no
electromagnetic radiation, and the greatest part of the remainder of this
book would have to be omitted. It was Maxwell's prediction that light
was an electromagnetic wave phenomenon, and that electromagnetic
waves of all frequencies could be produced, which drew the attention of all
physicists and stimulated so much theoretical and experimental research
into electromagnetism during the last part of the nineteenth century.
The set of four equations,
c c dt
V • B = VxE + — =
c dt
(6.28)
[Sect. 6.4] TimeVarying Fields, Maxwell's Equations, Conservation Laws 179
known as Maxwell's equations, forms the basis of all electromagnetic
phenomena. When combined with the Lorentz force equation and
Newton's second law of motion, these equations provide a complete
description of the classical dynamics of interacting charged particles and
electromagnetic fields (see Section 6.9 and Chapters 10 and 12). For
macroscopic media the dynamical response of the aggregates of atoms is
summarized in the constitutive relations which connect D and J with E,
and H with B (e.g., D = eE, J = <rE, B = /M for an isotropic, permeable,
conducting dielectric).
The units employed in writing Maxwell's equations (6.28) are those of
the previous chapters, namely, Gaussian. For the reader more at home in
other units, such as mks, Table 2 of the Appendix summarizes essential
equations in the commoner systems. Table 3 of the Appendix allows the
conversion of any equation from Gaussian to mks units, while Table 4
gives the corresponding conversions for given amounts of any variable.
6.4 Vector and Scalar Potentials
Maxwell's equations consist of a set of coupled firstorder partial
differential equations relating the various components of electric and
magnetic fields. They can be solved as they stand in simple situations.
But it is often convenient to introduce potentials, obtaining a smaller
number of secondorder equations, while satisfying some of Maxwell's
equations identically. We are already familiar with this concept in
electrostatics and magnetostatics, where we used the scalar potential <E> and
the vector potential A.
Since V • B = still holds, we can define B in terms of a vector potential :
B = V x A (6.29)
Then the other homogeneous equation in (6.28), Faraday's law, can be
written
V x (e + — ) =0 (6.30)
\ c dt!
This means that the quantity with vanishing curl in (6.30) can be written
as the gradient of some scalar function, namely, a scalar potential O :
c dt
c dt
(6.31)
180 Classical Electrodynamics
The definition of B and E in terms of the potentials A and <t> according to
(6.29) and (6.31) satisfies identically the two homogeneous Maxwell's
equations. The dynamic behavior of A and O will be determined by the
two inhomogeneous equations in (6.28).
At this stage it is convenient to restrict our considerations to the
microscopic form of Maxwell's equations. Then the inhomogeneous
equations in (6.28) can be written in terms of the potentials as
V 2 0> +   (V • A) = 4tt P (6.32)
cot
We have now reduced the set of four Maxwell's equations to two equations.
But they are still coupled equations. The uncoupling can be accomplished
by exploiting the arbitrariness involved in the definition of the potentials.
Since B is defined through (6.29) in terms of A, the vector potential is
arbitrary to the extent that the gradient of some scalar function A can be
added. Thus B is left unchanged by the transformation,
A * A' = A + VA (6.34)
In order that the electric field (6.31) be unchanged as well, the scalar
potential must be simultaneously transformed,
<D*<D' = <D_I^ (6.35)
c dt
The freedom implied by (6.34) and (6.35) means that we can choose a set
of potentials (A, <D) such that
1 dO
VA + — = (6.36)
c ot
This will uncouple the pair of equations (6.32) and (6.33) and leave two
inhomogeneous wave equations, one for O and one for A:
™Y^=4"P (637)
Equations (6.37) and (6.38), plus (6.36), form a set of equations equivalent
in all respects to Maxwell's equations.
[Sect. 6.5] TimeVarying Fields, Maxwell's Equations, Conservation Laws 181
6.5 Gauge Transformations; Lorentz Gauge; Coulomb Gauge
The transformation (6.34) and (6.35) is called a gauge transformation,
and the in variance of the fields under such transformations is called gauge
invar iance. The relation (6.36) between A and <I> is called the Lorentz
condition. To see that potentials can always be found to satisfy the
Lorentz condition, suppose that the potentials A, O which satisfy (6.32)
and (6.33) do not satisfy (6.36). Then let us make a gauge transformation
to potentials A', O' and demand that A', <!>' satisfy the Lorentz condition:
V.A' + i^OV.A + i^ + VAi^ (6.39)
c dt c dt c 2 dr
Thus, provided a gauge function A can be found to satisfy
the new potentials A', O' will satisfy the Lorentz condition and the wave
equations (6.37) and (6.38).
Even for potentials which satisfy the Lorentz condition (6.36) there is
arbitrariness. Evidently the restricted gauge transformation,
(6.41)
where V 2 A^^ = (6.42)
c 2 dr
preserves the Lorentz condition, provided A, $ satisfy it initially. All
potentials in this restricted class are said to belong to the Lorentz gauge.
The Lorentz gauge is commonly used, first because it leads to the wave
equations (6.37) and (6.38) which treat <D and A on equivalent footings,
and second because it is a concept which is independent of the coordinate
system chosen and so fits naturally into the considerations of special
relativity (see Section 11.9).
Another useful gauge for the potentials is the socalled Coulomb or
transverse gauge. This is the gauge in which
V • A = (6.43)
A^A + VA
^ ^ 13A
c dt
"
c*dt*
182 Classical Electrodynamics
From (6.32) we see that the scalar potential satisfies Poisson's equation,
V 2 0> = 4tt/) (6.44)
with solution,
0(x, i) = [ p(x '' ° d 3 x' (6.45)
J x — x'
The scalar potential is just the instantaneous Coulomb potential due to
the charge density />(x, t). This is the origin of the name "Coulomb
gauge."
The vector potential satisfies the inhomogeneous wave equation,
c 2 dt 2 c c dt
V 2 Aif=^J + ±V^ (6.46)
The "current" term involving the potential can, in principle, be calculated
from (6.45). Formally, we use the continuity equation to write
v ^ = _ v r^j(x^)^ {6A7)
dt J xx' v }
If the current is written as the sum of a longitudinal and transverse part,
J = J* + J, (6.48)
where V x J, = and V • J t = 0, then the parts can be written
J < = f v fr^i dV (6  49 >
477 J x — x I
J f =7 VxVx l^T i d * x ' ( 6  50 >
47T J x — x I
This can be proved by using the vector identity, V x (V x J) =
V(V • J)  V 2 J, together with V 2 (l/x  x') = 4tt <5(x  x'). Com
parison of (6.47) with (6.49) shows that
V^ = 4ttJ 1 (6.51)
at
Therefore the source for the wave equation for A can be expressed entirely
in terms of the transverse current (6.50):
V2A V^=J. (652)
 <T Ot ■ C
This is, of course, the origin of the name "transverse gauge."
[Sect. 6.6] TimeVarying Fields, Maxwell's Equations, Conservation Laws 183
The Coulomb or transverse gauge is often used when no sources are
present. Then 0> = 0, and A satisfies the homogeneous wave equation.
The fields are given by
13A1
E=  —
c dt
B = V x A
(6.53)
6.6 Green's Function for the TimeDependent Wave Equation
The wave equations (6.37), (6.38), and (6.52) all have the basic structure,
V>_iff=4 7 r/(x,0 (654)
r c 2 dt 2
where /(x, is a known source distribution. The factor c is the velocity
of propagation in the medium.
To solve (6.54) it is useful to find a Green's function for the equation,
just as in electrostatics. Since the time is involved, the Green's function
will depend on the variables (x, x\ t, t'), and will satisfy the equation,
( Vx 2  i ^G(x, t ; x', t') = 4tt d(x  x') b{t  t') (6.55)
Then in infinite space with no boundary surfaces the solution of (6.54)
will be
V<x, = Jg(x, t ; x', t')f(x', t') d*x' dt' (6.56)
Of course, the Green's function will have to satisfy certain boundary
conditions demanded by physical considerations.
The basic Green's function satisfying (6.55) is a function only of the
differences in coordinates (x — x') and times (t  t'). To find G we
consider the Fourier transform of both sides of (6.55). The delta functions
on the right have the representation,
<5(x  x') S(t  O = i 4 >fc fW k( ~V<> (6.57)
(2tt) J J
We therefore write the representation of G as
G(x, V, x', O = ja*k J^g(k, a>y k <~ V^<'> (6.58)
184 Classical Electrodynamics
The Fourier transform g(k, co) is to be determined. When (6 57) and
(6.58) are substituted into the defining equation (6.55), it turns out that
g(k, co) is
1 1
g(k, co) =
4tt 3
(* 2 ?)
(6.59)
When g(k, co) is substituted into (6.58) and the integrations over k and
co are begun, there appears a singularity in the integrand at k 2 = co 2 /c 2
Consequently solution (6.59) is meaningless without some rule as to how
to handle the singularities. The rule cannot come from the mathematics.
It must come from physical considerations. The Green's function satis
fying (6.55) represents the wave disturbance caused by a point source at x'
which is turned on only for an infinitesimal time interval at t' = t. We
know that such a wave disturbance propagates outwards as a spherically
diverging wave with a velocity c. Hence we demand that our solution for
G have the following properties :
(a) G = everywhere for t < t'.
(b) G represent outgoing waves for t > t' .
If we think of the co integration in (6.58), the singularities in^(k, co) occur
at co = ±ck. We can do the co integration as a Cauchy integral in the
complex co plane. For t > t' the integral along the real axis in (6.58) is
equivalent to the contour integral around a path C closed in the lower
halfplane, since the contribution on the semicircle at infinity vanishes
exponentially. On the other hand, for t' > t, the contour must be closed
in the upper halfplane, as shown in Fig. 6.4 by path C.
In order to make G vanish for t < t' we must imagine that the poles at
co = ±ck are displaced below the real axis, as in Fig. 6.4. Then the integral
over C for t > t' will give a nonvanishing contribution, while the integral
t<t'
Fig. 6.4 Complex to plane with contour
C for / > /' and contour C for t < t'.
[Sect. 6.6] TimeVarying Fields, Maxwell's Equations, Conservation Laws 185
over C for t < t' will vanish. The displacement of the poles can be
accomplished mathematically by writing (w + ie) in place of a) in (6.59).
Then the Green's function is given by
/. /• ikR ia>T
G(x, t; x\ t') = — I d*k\ da> e  (6 M)
c 2
where R = x — x', t = t — t', and e is a positive infinitesimal.
The integration over co for t > can be done with Cauchy's theorem
applied around the contour C of Fig. 6.4, giving
G = JL(a* k j**™LlciV (6.61)
2tt*J k
The integration over d z k can be accomplished by first integrating over
angles. Then
G = — \dk sin (kR) sin (cr/c) (6.62)
Since the integrand is even in k, the integral can be written over the whole
interval, — oo < k < oo. With a change of variable x = ck, (6.62) can
be written
G = J— f °° <te(e i[r " (B/c)]a! — e i[r+(B/c)]a! ) (6.63)
From (2.52) we see that the integrals are just Dirac delta functions. The
argument of the second one never vanishes (remember, t > 0). Hence
only the first integral contributes, and the Green's function is
or, more explicitly,
G(x, v, x', O = ^— : , " < 6  64 >
x  x'
This Green's function is sometimes called the retarded Green's function
because it exhibits the causal behavior associated with a wave disturbance.
The effect observed at the point x at time t is due to a disturbance which
x  x' . . .
originated at an earlier or retarded time t' = t at the point x .
186 Classical Electrodynamics
The solution for the wave equation (6.54) in the absence of boundaries
is
ip(x, t) =
y c Lfv.ndv* (6  65)
X — X
The integration over dt' can be performed to yield the socalled "retarded
solution,"
^(x,0 = f^^% t dV (6.66)
J x — x 
The square bracket [ ] ret means that the time t' is to be evaluated at the
retarded time, t' = t — * .
6.7 InitialValue Problem; Kirchhoff's SurfaceIntegral Representation
Solution (6.66) is a particular integral of the inhomogeneous wave
equation (6.54). To it can be added any solution of the homogeneous
wave equation necessary to satisfy the boundary conditions. From the
table at the end of Section 1.9 we see that the proper boundary conditions
are Cauchy boundary conditions (ip and dy/dn given) on an "open surface."
For the threedimensional wave equation an open surface is defined as a
threedimensional volume specified by one functional relationship between
the four coordinates (x, y, z, t). The customary open surface is ordinary
threedimensional space at a fixed time, t = t . Then the problem is an
initialvalue problem with y>(x, t ) and dy(x, t )/dt given for all x. We
wish to determine ip(x, i) for all times t > t .
To discuss the initialvalue problem and also an integral representation
of Kirchhoff for closed bounding surfaces, we use Green's theorem (1.35),
integrated over time from t' = t to t' = t x > t:
[ V [ <*V(0 V V  V V V) = f V iL%L v *±) da' (6.67)
We now choose yj = y> and <f> = G. With wave equations (6.54) and (6.55)
the lefthand side can be written
LHS =J l dt'j dVUrrvKx', t') d(x'  x) d(t'  t)
 W .OC + i(ejirg)] (6.68)
[Sect. 6.7] TimeVarying Fields, Maxwell's Equations, Conservation Laws 187
The first two terms in (6.68) will evidently give the particular integral (6.66).
The last two terms can be integrated by parts with respect to the time to
give
LHS = 4TTtp(x, t)  4tt j V j dV/(x\ t')G
c 2 Jr \ dt' dt r=t
Since G = at t' = t x > t, the upper limit vanishes. We can thus combine
(6.69) and (6.67) to give the integral representation for y(x, i) inside the
volume V, bounded by the surface S, at times t> t :
We have written the first term in (6.70) in the usual form (6.66) by using
the explicit result (6.64) for G. We now do the same for the other terms.
For simplicity, first we consider the infinitedomain initialvalue
problem with xp and dxpfdt as given functions of space at t = t = 0:
V,(x,0) = F(x), ^(x,0)=D(x)
at
(6.71)
Then the surface integral in (6.70) can be omitted. To simplify the
notation we take the observation point at the origin and use spherical
coordinates in the integrals. Then we have
[D(/ >Q o4') F(r '> Q) l4' + 7 r )L
X
The derivative of the delta function can be written
= c 2 d'(r'  ct)
(6.72)
MH
t'=0
Then, with the properties of the delta function summarized in Section 1.2,
(6.72) becomes
Y<0, = fdQ' J*°V dr'f[r', a',t' = t^J
+ L La[tD(ct, a r ) + 1 (tF(ct, Q'»
(6.73)
188 Classical Electrodynamics
This is called PoissorCs solution to the initialvalue problem. With no
sources present (/= 0), only values of the initial field at distances ct from
the origin contribute at time t.
The initialvalue problem for the wave equation has been extensively
studied in one, two, three, and more dimensions. The reader is referred to
Morse and Feshbach, pp. 843847, and to the more mathematical treat
ment of Hadamard.
The other result which we wish to obtain from (6.70) is the socalled
Kirchhoff representation of the field inside the volume V in terms of the
values of %p and its derivatives on the boundary surface S. We thus assume
that there are no sources within V and that the initial values of y and
dxpjdt vanish. (Alternatively, we can assume that the initial time is in the
remote past so that there are no more contributions from the initialvalue
solution (6.73) within the volume V.) Then the field inside V is given by
y>(x, = J f V £> da'io ^  y, *2) (6.74)
47rJt Js \ dn' Y dn'J K J
With G given by (6.64) we can calculate dG/dn' :
dR RdR
R
.( ^H , 4+H )
*l R 2 cR I
(6.75)
The term involving the derivative of the delta function can be integrated
by parts with respect to the time t' . Then the Kirchhoff integral repre
sentation is
AttJs I R R 3 * y J C R % dt' Jret
(6.76)
where R = x — x', and n is a unit normal to the surface S. We emphasize
that (6.76) is not a solution for the field xp, but only an integral representa
tion in terms of its value and the values of its space and time derivatives on
the surface. These cannot be specified arbitrarily; they are known only
when the appropriate Cauchy boundaryvalue problem has been solved.
The Kirchhoff integral (6.76) is a mathematical statement of Huygens'
principle and is used as the starting point in discussing opticaldiffraction
problems. Diffraction is discussed in detail in Chapter 9, Section 9.5 and
below.
[Sect. 6.8] TimeVarying Fields, Maxwell's Equations, Conservation Laws 189
6.8 Poynting's Theorem
The forms of the laws of conservation of energy and momentum are
important results to establish for the electromagnetic field. We begin by
considering conservation of energy, often called Poynting's theorem (1884).
For a single charge q the rate of doing work by external electromagnetic
fields E and B is qv • E, where v is the velocity of the charge. The magnetic
field does no work, since the magnetic force is perpendicular to the velocity.
If there exists a continuous distribution of charge and current, the total
rate of doing work by the fields in a finite volume V is
f J.Ed 3 * (6.77)
Jv
This power represents a conversion of electromagnetic energy into
mechanical or thermal energy. It must be balanced by a corresponding
rate of decrease of energy in the electromagnetic field within the volume V.
In order to exhibit this conservation law explicitly, we will use Maxwell's
equations to express (6.77) in other terms. Thus we use the Ampere
Maxwell law to eliminate J:
f J.Ed 3 x = f [cE.(VxH)E.^l^ (6.78)
J v AttJv L ot J
If we now employ the vector identity,
V • (E x H) = H • (V x E)  E • (V x H)
and use Faraday's law, the right side of (6.78) becomes
l j  E ^=^H cv  (ExH)+E f +H f
d 3 x (6.79)
To proceed further we must make two assumptions. The first one is not
fundamental, and is made for simplicity only. We assume that the macro
scopic medium involved is linear in its electric and magnetic properties.
Then the two time derivatives in (6.79) can be interpreted, according to
equations (4.92) and (6.16), as the time derivatives of the electrostatic and
magnetic energy densities. We now make our second assumption, namely,
that the sum of (4.92) and (6.16) represents the total electromagnetic
energy, even for timevarying fields. Then if the total energy density is
denoted by
u = — (E • D + B • H) (6.80)
190 Classical Electrodynamics
(6.79) can be written
f J.Ed 3 *=f [^ + lv.(ExH)
JV JV Lot 477
d 3 x (6.81)
Since the volume V is arbitrary, this can be cast into the form of a dif
ferential continuity equation or conservation law,
^ + VS=J.E (6.82)
The vector S, representing energy flow, is called Poynting's vector. It is
given by
S = .JL ( E x H) (6.83)
47T
and has the dimensions of (energy/area x time). Since only its divergence
appears in the conservation law, Poynting's vector is arbitrary to the
extent that the curl of any vector field can be added to it. Such an added
term can, however, have no physical consequences. Hence it is customary
to make the specific choice (6.83).
The physical meaning of the integral or differential form (6.81) or (6.82)
is that the time rate of change of electromagnetic energy within a certain
volume, plus the energy flowing out through the boundary surfaces of the
volume per unit time, is equal to the negative of the total work done by the
fields on the sources within the volume. This is the statement of conser
vation of energy. If nonlinear effects, such as hysteresis in ferromagnetic
materials, are envisioned, the simple law (6.82) is no longer valid, but
must be supplemented by terms giving the hysteresis power loss.
6.9 Conservation Laws for a System of Charged Particles
and Electromagnetic Fields
The statements (6.81) and (6.82) of Poynting's theorem have empha
sized the energy of the electromagnetic fields. The work done per unit time
per unit volume by the fields (J • E) is a conversion of electromagnetic into
mechanical or heat energy. Since matter is ultimately composed of
charged particles (electrons and atomic nuclei), we can think of this rate
of conversion as a rate of increase of energy of the charged particles per unit
volume. Then we can interpret Poynting's theorem for the microscopic
fields as a statement of conservation of energy of the combined system of
particles and fields. If we denote the total energy of the particles within
[Sect. 6.9] Time Varying Fields, Maxwell's Equations, Conservation Laws 191
the volume V as E mech and assume that no particles move out of the
volume, we have ,.
™2™* = \ J.EcPx (6.84)
dt Jv
Then Poynting's theorem expresses the conservation of energy for the
combined system as
^ = 4 ( £ mech + Efleld) =  <P n • S da (6.85)
dt dt Js
where the total field energy within V is
£ fl eid = I u d z x = — I (E 2 + B 2 ) d z x (6.86)
Jv SttJv
The conservation of linear momentum can be similarly considered. We
have seen that the force on a charge q in an external field E is qE. From the
basic law (5.12) for forces on currents we can deduce that the magnetic
force on a charge q moving with velocity v in an external magnetic induction
B is (q/c) v x B. Thus the total electromagnetic force on a charged particle
18 / v \
F=tf(E + xBl (6.87)
This is called the Lor entz force. Although we have deduced it within the
framework of steadystate phenomena, it is well verified for all charged
particles with arbitrarily large velocities.
From Newton's second law we can write the rate of change of the
particle's momentum as
jr q ( E+ l* B ) (6  88)
If the sum of all the momenta of all the particles in the volume V is
denoted by P mech , we can write
= ( P E +ij x B)d 3 x (6.89)
Jv c
^Pmech _ j (^ j_ 1 T v BU3,
dt
We have converted the sum over particles to an integral over charge and
current densities for convenience in manipulation. The particulate nature
can be recovered at any stage by making use of delta functions, as in
Section 1.2. In the same manner as for Poynting's theorem, we use
Maxwell's equations to eliminate p and J from (6.89):
4tt 4tt\ c dt/
(6.90)
192 Classical Electrodynamics
Note that we have written only E and B in (6.90), and not H or D. The
reason is, as mentioned earlier, that we are imagining all the charges as
treated in the mechanical part of the system and so use the microscopic
equations which involve only E and B. Some remarks will be made in
the next section on the differences which arise when some of the particles,
namely, the bound atoms, are included in the "field" energy and momentum
through the dielectric constant and permeability. (See also Problem 6.8.)
With (6.90) substituted into (6.89) the integrand becomes
pE + JxB = —
1 /5F
E(V • E) +  B x —  B x (V x B)
(6.91)
c Att\ c dt
Then writing
Bx^=i( E xB) + Ex?5
dt dt } dt
and adding B(V • B) = to the square bracket, we obtain
P E +  J x B = — [E(VE) + B(VB)  E x (V x E)
c 4n
 B x (V x B)]  —  (E x B) (6.92)
4rrcdt
The rate of change of mechanical momentum (6.89) can now be written
^JES* + ± f J_( E XB)A = 1( [E(VE)  E x (V x E)
dt dtJv47TC 4ttJv
+ B(V • B)  B x (V x B)] d*x (6.93)
We may tentatively identify the volume integral on the left as the total
electromagnetic momentum P fleld in the volume V:
field
= — (E x B) d 3 x (6.94)
AttcJv
The integrand can be interpreted as a density of electromagnetic
momentum. We note that this momentum density is proportional to the
energyflux density S, with proportionality constant c~ 2 . ,
To complete the identification of the volume integral of — (E x B) as
electromagnetic momentum, and to establish (6.93) as the conservation
law for momentum, we must convert the volume integral on the right into
a surface integral of the normal component of something which can be
identified as momentum flow.
[Sect. 6.9] TimeVarying Fields, Maxwell's Equations, Conservation Laws 193
Evidently the terms in the volume integral (6.93) transform like vectors.
Consequently, if they are to be combined into the divergence of some
quantity, that quantity must be a tensor of the second rank. While it is
possible to deal with rectangular components of momentum, instead of the
vectorial form (6.93), the tensor can be handled within the framework of
vector operations by introducing a corresponding dyadic. If a tensor in
three dimensions is denoted by T u (i,j =1,2, 3), and e* are the unit base
vectors of the coordinate axes, the dyadic corresponding to the tensor T i}
is defined to be
Y= le^e. (6.95)
t=l 1 = 1
The unit vector on the left can form scalar or vector products from the
left, and correspondingly for the unit vector on the right. Given the dyadic,
we can determine the tensor elements by taking the appropriate scalar
products :
r„ = e, V. c, (6.96)
A special dyadic is the identity I formed with the unit secondrank tensor:
<— >
I = € x € x + e 2 € 2 + £,€3 (6.97)
<>
The scalar product of any vector or vector operation with I from either
the left or right merely gives the original vector quantity.
With these sketchy remarks about dyadics, we now proceed with the
vector manipulations needed to convert the volume integral on the right
side of (6.93) into a surface integral. Using the vector identity,
£V(B • B) = (B • V)B + B x (V x B)
the terms involving B in (6.93) can be written
B(V • B)  B x (V x B) = B(V • B) + (B • V)B  %VB 2 (6.98)
This can be identified as the divergence of a dyadic:
B(V . B) + (B • V)B  %VB 2 = V • (BB  YlB 2 ) (6.99)
The electric field term in (6.93) can be put in this same form. Consequently
the conservation of linear momentum becomes
T (Pmech + Pfleld) =  V Yd 8 * = in .? da (6.100)
dt Jv Js
<— >
The tensordyadic T, called Maxwell's stress tensor, is
Y = — [EE + BB  \\(E 2 + fl 2 )] (6.101)
Att
194 Classical Electrodynamics
The elements of the tensor are
T it = J \E t E, + B t B t  tfifp + fl 2 )] (6.102)
477
Evidently (— n • T) in (6.100) represents the normal flow of momentum
per unit area out of the volume V through the surface S. Or, in other
words, (— n • T) is the force per unit area transmitted across surface S.
This can be used to calculate the forces acting on material objects in
electromagnetic fields by enclosing the objects with a boundary surface S
and adding up the total electromagnetic force according to the righthand
side of (6. 100).
The conservation of angular momentum of the combined system of
particles and fields can be treated in the same way as we have handled
energy and linear momentum. This is left as a problem for the student
(Problem 6.9).
6.10 Macroscopic Equations
Although the equations of electrodynamics have been written in macro
scopic form for the most part in this chapter, the reader will be aware that
the derivation of the macroscopic equations from the microscopic ones
was done separately for electro statics and magnetostatics in Sections 4.3
and 5.8. Thus there arises the question of whether the derivation still holds
good for timedependent fields. It is intuitively obvious that it must, since
Maxwell's addition of the displacement current was done at the macro
scopic level. Nevertheless, it is useful to examine briefly the derivation to
see in particular how the time variation of the polarization P gives rise to
a current contribution and so converts the microscopic displacement
current dE/dt into the macroscopic displacement current dD/dt.
The basic assumption inherent in our previous discussions was that the
macroscopic fields E and B which satisfy the two homogeneous Maxwell's
equations (6.28) are the averages of the corresponding microscopic fields
e and (3 :
E(x,0=<€>, B(x,0 = <P> (6.103)
The averages now involve a temporal and a spatial average, e.g.,
<e> = g±^ jd*tjdTt(x + 5, t + t) (6104)
where the volume AK and the time interval Ar are small compared to
macroscopic quantities.
[Sect. 6.10] TimeVarying Fields, Maxwell's Equations, Conservation Laws 195
Relations (6.103) imply that the macroscopic potentials C> and A are the
averages of their microscopic counterparts,
0>(x, /) = <<£>, A(x,0=<a> (6.105)
since the fields E and B (or e and (5) are derived by differentiation according
to (6.29) and (6.31).
The derivation of the averaged potentials in terms of the molecular
properties proceeds exactly as in Sections 4.3 and 5.8, with two modifi
cations. The first is that, according to our discussion of the solution of the
wave equation in Section 6.6, we must have a "retarded" solution. Thus the
same steps that led to (4.33) now lead to the averaged scalar potential:
(<f>) = ( N (x')\^^P + < Pmo i(x', O) ' W— L ^t)1 **
J L x — X  \X — x/Jret
(6.106)
To this must be added the retarded contribution of the excess free charge,
p cx . The second change comes in the vector potential. In the steadystate
situation the molecular contribution to the vector potential was the sum of
terms like (5.75), representing magnetic dipole contributions. The leading
term in the expansion vanished because of the condition V • J = 0. With
timedependent fields this is no longer true. If we retrace our steps to
equation (5.51), we see that the leading term in the expansion of a mol is
~ ~ * = * I*"' ° " + ("te™^) 016 ) + • • • < 61 ° 7 >
For simplicity we omit the retarded symbols temporarily. Using the
identity V • (x/J) = J t + x/V . J, the first term can be transformed into
a mo i = —, x'(V' • J mo i) d 3 x' + (5.75) +
ex — xj J
(6.108)
The continuity equation can now be used to write V • J mol = —dp mo Jdt',
and the definition of molecular electric dipole moment (4.25) can be
employed to cast (6.108) in the form:
a mo i(x, t)
j dp 3 (0 1 + m,.(Q x (x  x 3 )"
(6.109)
C dt' X — X, ' X — X, 3 J ret
With timedependent fields we have a leading term proportional to the
time rate of change of the electric dipole moment.* Summing over all
* Actually, for timevarying fields not only does the leading term appear, but also from
3Q / 1 \ <>
the second term in (5.51) there arises a term which is — — • V I • I , where Q
c dt \x  x /
is the quadrupole dyadic of the molecule. Since we kept only electric dipole terms in
(6.106), we drop this quadrupole term here.
196
Classical Electrodynamics
molecules and averaging according to (6.104) leads to the averaged
microscopic vector potential:
(a)
')>
= ifjV(x')r^;<Pmoi(x',n
c J Lx — x  Ot
+ c<m mo i(x', i')> x W , * ) dV (6.110)
\ X — X / Jret
To this must be added the standard contribution from the macroscopic
conductioncurrent density J(x, *)•
Solutions (6.106) and (6.110), augmented by the freecharge and con
ductioncurrent contributions, can be written as
<*>Jfc^L" 1
cJ L x — x
(6.111)
With definitions (4.36) and (5.77) of macroscopic polarization P and
magnetization M, the averaged charge and current densities in (6.111) can
be expressed as
(p) = pVP
ot
where p and J are the macroscopic charge and current densities.
We are now in a position to verify explicitly the deduction of the
macroscopic Maxwell's equations from the microscopic ones. The
homogeneous ones follow directly from identification (6.103). Of the
inhomogeneous ones, consider the microscopic form of Ampere's law:
_ _ 4t7 t , 13c
c c ot
(6.113)
Averaging both sides and using (6.112) for <J>, we get
4tt / „ 3P\ 1 9E tr „ „ x
VxB = T (j + c(V*M) + ^)+ F (6.114)
With the definitions H = B — 47rM and D = E + 4ttP, this becomes
(6.115)
_ TT 4tt t , ldD
V x H = — J +  —
c c ot
as required. The other equation, V • D = Airp, follows in an even simpler
way from (6.112).
[Sect. 6.10] TimeVarying Fields, Maxwell's Equations, Conservation Laws 197
As a final remark concerning the macroscopic field equations we discuss
the differences between the microscopic and macroscopic forms of
Poynting's theorem. We derived the conservation of energy in Section
6.8 in the macroscopic form (6.81). Written out explicitly in terms of all
the fields, it is
^f (E x H).n«fa + f (e.^ + H—W*^ _f E.Jd*x
Air J s AttJv\ ot at/ Jv
(6.116)
The different fields E, D, B, H enter in characteristic ways which can be
understood if we establish contact with the microscopic form of Poynting's
theorem. We can do this most easily by merely expressing the left side of
(6.116) in terms of the basic fields E and B. Then (6.116) can easily be
shown to be
^((ExB).nrfa + i[ ( E • ^ + B . ??) d*x
AttJs AttJv \ at ot I
=  I E •( J + cV x M + — j d z x (6.117)
From (6.112) we see that (6.117) looks like the statement of Poynting's
theorem for the microscopic fields, except that each quantity is replaced by
its average. This is not the average of Poynting's theorem for microscopic
fields, but differs from it by a set of terms which are the statement of energy
conservation for the fluctuating fields measuring the instantaneous
departure of € and (3 from E and B. Apart from these fluctuating fields,
(6.117) can be interpreted as follows.
If we include in the sources of charge and current the electronic motion
within the molecules as well as the conduction current, then Poynting's
theorem appears in terms of the basic fields E and B and involves the work
done per unit time by the electric field on all currents. If we choose to
rap 1
treat the work done on the effective molecular current — + c(V x M)
Ot
as energy stored or propagated in the medium, that term can be taken
over to the lefthand side and included in the energydensity and energy
flow terms characteristic of the medium. Then we return to the macro
scopic Poynting's theorem (6.116) with only the work done per unit time
by the electric field on the conduction current shown explicitly. It is
natural to absorb the energy associated with the effective molecular
current into the energy stored in the field, since it is a property of the
medium and is in general stored energy (i.e., reactive power) which involves
no timeaverage dissipation (not true for magnetic media with hysteresis
198 Classical Electrodynamics
effects). The power associated with the conduction current is, on the
other hand, dissipative, since it involves a conversion of electrical energy
into mechanical.
REFERENCES AND SUGGESTED READING
The conservation laws for the energy and momentum of electromagnetic fields are
discussed in almost all textbooks. A good treatment of the energy of quasistationary
currents and forces acting on circuits carrying currents, different from ours, is given by
Panofsky and Phillips, Chapter 10.
Stratton, Chapter II,
discusses the Maxwell stress tensor in some detail in considering forces in fluids and
solids. The general topics of conservation laws, as well as quasistationary circuit theory,
inductance calculations, and forces, are dealt with in a lucid manner by
Abraham and Becker, Band I, Chapters VIII and IX.
Inductance calculations and circuit theory are also presented by
Smythe, Chapters VIII, IX, and X,
and many engineering textbooks.
The hereneglected subject of eddy currents and induction heating is discussed with
many examples by
Smythe, Chapter XI.
The mathematical topics in this chapter center around the wave equation. The
initialvalue problem in one, two, three, and more dimenisons is discussed by
Morse and Feshbach, pp. 843847,
and, in more mathematical detail, by
Hadamard.
PROBLEMS
6.1 (a) Show that for a system of currentcarrying elements in empty space the
total energy in the magnetic field is
w  .,,...,^J(xW(x')
= h*r x y x '
where J(x) is the current density.
(b) If the current configuration consists of n circuits carrying currents
I x , I 2 , ■ ■ ■ , /„, show that the energy can be expressed as
n n n
W = \ I Ul? +2 2 Miihh
i=l i=\ j>i
Exhibit integral expressions for the selfinductances (L 2 ) and the mutual
inductances (M i3 ).
6.2 A twowire transmission line consists of a pair of nonpermeable parallel
wires of radii a and b separated by a distance d > a + b, A current flows
[Probs. 6] Time Varying Fields, Maxwell's Equations, Conservation Laws 199
down one wire and back the other. It is uniformly distributed over the
cross section of each wire. Show that the selfinductance per unit length is
6.3
6.4
6.5
c 2 L = 1 + 2 In
©
A circuit consists of a thin conducting shell of radius a and a parallel
return wire of radius b inside. If the current is assumed distributed
uniformly throughout the cross section of the wire, calculate the self
inductance per unit length. What is the selfinductance if the inner con
ductor is a thin hollow tube ?
Show that the mutual inductance of two circular coaxial loops in a homo
geneous medium of permeability /* is
M 12 = ^ Vab
where
[(H
(!*)*<*)!**)
k 2 =
Aab
(a + bf + d 2
and a, b are the radii of the loops, d is the distance between their centers,
and K and E are the complete elliptic integrals.
Find the limiting value when d < a, 6 and a ~ b.
A transmission line consists of two, parallel perfect conductors of arbitrary,
but constant, cross section. Current flows down one conductor and returns
via the other.
Show that the product of the inductance per unit length L and the
capacitance per unit length C is
LC =
fi€
where /a, and e are the permeability and the dielectric constant of the medium
surrounding the conductors, while c is the velocity of light in vacuo.
6.6 Prove that any vector field F can be decomposed into transverse and
longitudinal parts,
F = F £ + ¥ t
with VF ( =OandVxF ( =0, where F { and F t are given by (6.49) and
(6.50).
200 Classical Electrodynamics
6.7 (a) Show that the onedimensional wave equation,
d 2 ip 1 d^xp
~fa?~c*~di* == °
has the general solution,
I x\ I x\ c C t+ix/c)
where the boundary conditions are specified by the values of y> and dy>/ dx
at x = for all time:
V'(0,0=/(0, d £(0,t)=F(t)
(b) What is the corresponding solution if the boundary conditions are
that, at t = 0,
y>(x,Q)=f{x), ?l(x,0)=g(x)
6.8 Discuss the conservation of energy and linear momentum for a macro
scopic system of sources and electromagnetic fields in a medium described
by a dielectric constant e and a permeability fi. Show that the energy
density, Poynting's vector, fieldmomentum density, and Maxwell stress
tensor are given by
u = i (eE 2 + fxH 2 )
OTT
S = £ (E x H)
g =£I(ExH)
'rTT
What modifications arise if e and fj, are functions of position ?
6.9 With the same assumptions as in Problem 6.8 discuss the conservation of
angular momentum. Show that the differential and integral forms of the
conservation law are
and
r\ 4 v
Y t (^mech + Afield) + V • M =
j\ (J^mech + Afield) d 3 X + \ n • M da =
where the field angularmomentum density is
J^fleld = XXg=£^XX(ExH)
[Probs. 6] TimeVarying Fields, MaxwelVs Equations, Conservation Laws 201
and the flux of angular momentum is described by the tensor
M = T x x
<— >•
Note: M can be written as a thirdrank tensor, M m = T f jX k — r,^.
But in the indices j and it it is antisymmetric and so has only three inde
pendent elements. Including the index i, M m therefore has nine components
and can be written as a pseudo tensor of the second rank, as above.
6.10 A plane wave is incident normally on a perfectly absorbing flat screen.
(a) From the law of conservation of linear momentum show that the
pressure (called radiation pressure) exerted on the screen is equal to the
field energy per unit volume in the wave.
(b) In the neighborhood of the earth the flux of electromagnetic energy
from the sun is approximately 0.14 watt/cm 2 . If an interplanetary "sail
plane" had a sail of mass 10~ 4 gm/cm 2 of area and negligible other weight,
what would be its maximum acceleration in centimeters per square second
due to the solar radiation pressure? How does this compare with the
acceleration due to the solar "wind" (corpuscular radiation)?
6.11 A circularly polarized plane wave moving in the z direction has a finite
extent in the x and y directions. Assuming that the amplitude modulation
is slowly varying (the wave is many wavelengths broad), show that the
electric and magnetic fields are given approximately by
r i(*E a dE a \
E(x, y, z, t) =* \E (x, yfa ± fc») + j\~^ ± i ~^J^
gikz—icot
B=;T iVyue E
where e x , e 2 , e 3 are unit vectors in the x, y, z directions.
6.12 For the circularly polarized wave of Problem 6.11 calculate the time
averaged component of angular momentum parallel to the direction of
propagation. Show that the ratio of this component of angular momentum
to the energy of the wave is
u
Interpret this result in terms of quanta of radiation (photons). Show that
for a cylindrically symmetric, finite plane wave the transverse components
of angular momentum vanish.
7
Plane Electromagnetic Waves
This chapter is concerned with plane waves in unbounded, or
perhaps semif infinite, media. The basic properties of plane waves in non
conducting media — their transverse nature, the various states of polari
zation — are treated first. Then the behavior of onedimensional wave
packets is discussed; group velocity is introduced; dispersive effects are
considered. Reflection and refraction of radiation at a plane interface
between dielectrics are presented. Then plane waves in a conducting
medium are described, and a simple model of electrical conductivity is
discussed. Finally the conductivity model is modified to apply to a
tenuous plasma, or electron gas, and the propagation of transverse waves
in a plasma in the presence of an external static magnetic field is con
sidered.
7.1 Plane Waves in a Nonconducting Medium
A basic feature of Maxwell's equations for the electromagnetic field is
the existence of traveling wave solutions which represent the transport of
energy from one point to another. The simplest and most fundamental
electromagnetic waves are transverse, plane waves. We proceed to see how
such solutions can be obtained in simple nonconducting media described
by spatially constant permeability and susceptibility. In the absence of
sources, Maxwell's equations in an infinite medium are :
VE = V x E + — =0
c dt
VB = VxB^^ =
c dt
202
(7.1)
[Sect. 7.1] Plane Electromagnetic Waves 203
where the medium is characterized by the parameters fi, e. By combining
the two curl equations and making use of the vanishing divergences, we
find easily that each cartesian component of E and B satisfies the wave
equation:
V2 "I?i = (7  2)
v 2 dt 2
where
_ c
(7.3)
is a constant of the dimensions of velocity characteristic of the medium.
The wave equation (7.2) has the wellknown planewave solutions:
kxi<ot /n ^\
u = e
where the frequency co and the magnitude of the wave vector k are related
by
CO , — CO
fc = ^ = V^ (7.5)
V c
If we consider waves propagating in only one direction, say, the x
direction, the fundamental solution is
u(x, i) = Ae ikx  iat + Be ikx  imt (7.6)
Using (7.5), this can be written
u k (x, t) = Ae ik <* Vt) + Be~ mx+Vt) (7.7)
If v is not a function of A: (i.e., a nondispersive medium, with fxe independent
of frequency), we know by the Fourier integral theorem (2.50) and (2.51)
that by linear superposition we can construct from u k (x, t) a general
solution of the form:
u (x, t) =f(x  vt) + g(x + vt) (7.8)
where f(z) and g(z) are arbitrary functions. It is easy to verify directly
that this is a solution of the wave equation (7.2). Equation (7.8) represents
waves traveling to the right and to the left with velocities of propagation
equal to v, which is called the phase velocity of the wave. If v is a function
of k, the situation is not as simple— the initial waves f(x) and g(x) are not
propagated without distortion at velocity v (see Section 7.3). For each
frequency component, however, v given by (7.3) is still the phase velocity.
The basic plane wave (7.4) and (7.5) satisfies the scalarwave equation
(7.2). But we still must consider the vector nature of the electromagnetic
fields and the requirement of satisfying Maxwell's equations. With the
convention that the physical electric and magnetic fields are obtained by
204
Classical Electrodynamics
taking the real parts of complex quantities, we assume that the plane
wave fields are of the form :
E(x,t) = e 1 E e** ia *
ik • x—i<ot
(7.9)
B(x, = ejV
where c l5 c 2 are two constant real unit vectors, and E , B are complex
amplitudes which are constant in space and time. The requirements
V • E = and V • B = demand that
€l • k = 0,
c 2 • k =
(7.10)
This means that E and B are both perpendicular to the direction of
propagation k. Such a wave is called a transverse wave. The curl equations
provide further restrictions. Substitution of (7.9) into the first curl
equation in (7.1) leads to the relation:
(k x €!)£ €380
ik • x— i<"t
=
Equation (7.11) (really several equations) has the solution:
€„ =
kxc t
and
B = V^e E
(7.11)
(7.12)
(7.13)
This shows that (c l5 c 2 , k) form a set of orthogonal vectors and that E and
B are in phase and in constant ratio, as indicated in Fig. 7.1. The wave
described by (7.9), (7.12), and (7.13) is a transverse wave propagating in
the direction k. It represents a timeaveraged flux of energy given by the
Fig. 7.1 Propagation vector k and
two orthogonal polarization vectors
6i and € 8 .
[Sect. 7.2] Plane Electromagnetic Waves 205
real part of the complex Poynting's vector:
S = — E x H* (7.14)
2 4tt
The energy flow (energy per unit area per unit time) is
S = ^ /^£  2 € 3 (7.15)
J 0\ «3
87T 'V fl
where €3 is a unit vector in the direction of k. The timeaveraged density
u is correspondingly
u = — (cE • E* + B • B*) (7.16)
l&TT fl
This gives
M = f£ol 2 (717)
The ratio of the magnitude of (7.15) to (7.17) shows that the velocity of
energy flow is v = clV/txe, as expected from (7.8).
7.2 Linear and Circular Polarization
The plane wave (7.9) is a wave with its electric field vector always in the
direction e v Such a wave is said to be linearly polarized with polarization
vector € v To describe a general state of polarization we need another
linearly polarized wave which is independent of the first. Clearly the two
waves
ik • x— io>t
with B, = y/fte —  — ', j = 1,2
(7.18)
represent two such linearly independent solutions. The amplitudes E x
and E 2 are complex numbers to allow the possibility of a phase difference
between the waves. A general solution for a plane wave propagating in
the direction k is given by a linear combination of E x and E 2 :
E(x, = (eA + c^y* * x  i(ot (7.19)
If E x and E 2 have the same phase, (7.19) represents a linearly polarized
wave, with its polarization vector making an angle 6 = tan 1 (EJE^) with
c x and a magnitude E = Ve^ + E 2 2 , as shown in Fig. 7.2.
If E x andE 2 have different phases, the wave (7.19) is elliptically polarized.
Classical Electrodynamics
E
Fig. 7.2 Electric field of a linearly polarized
wave.
To understand what this means let us consider the simplest case, circular
polarization. Then E x and E 2 have the same magnitude, but differ in phase
by 90°. The wave (7.19) becomes:
E(x, = £ ( ei ± /e 2 y' k • %  iait (7.20)
with E the common real amplitude. We imagine axes chosen so that the
wave is propagating in the positive z direction, while e x and € 2 are in the x
and y directions, respectively. Then the components of the actual electric
field, obtained by taking the real part of (7.20), are
(7.21)
EJx, t) = E cos (kz — cut)
E y (x, t) = TE sin (kz — wt)
At a fixed point in space, the fields (7.21) are such that the electric vector
is constant in magnitude, but sweeps around in a circle at a frequency co,
as shown in Fig. 7.3. For the upper sign (e x + /e 2 ), the rotation is counter
clockwise when the observer is facing into the oncoming wave. This wave
is called left circularly polarized in optics. In the terminology of modern
physics, however, such a wave is said to have positive helicity. The latter
description seems more appropriate because such a wave has a positive
projection of angular momentum on the z axis (see Problem 6.12). For
the lower sign (c x — /e 2 ), the rotation of E is clockwise when looking into
E(x, = i5o(ci + i«2)«'
Fig. 7.3 Electric field of a circularly polarized
wave.
[Sect. 7.2]
Plane Electromagnetic Waves
207
Fig. 7.4 Electric field and magnetic induction for an elliptically polarized wave.
the wave; the wave is right circularly polarized (optics); it has negative
helicity.
The two circularly polarized waves (7.20) form an equally acceptable
set of basic fields for description of a general state of polarization. We
introduce the complex orthogonal unit vectors :
with properties
c± = 7= (ci ± ie 2 )
(7.22)
(7.23)
(7.24)
e ± *.€ T =0
e±*e 3 =
€ ± *.€ ± =1
Then a general representation, equivalent to (7.19), is
E(x, t) = (£+€+ + E_€_)e
where E+ and E_ are complex amplitudes. If E + and E_ have different
magnitudes, but the same phase, (7.24) represents an elliptically polarized
wave with principal axes of the ellipse in the directions of e x and c 2 . The
ratio of semimajor to semiminor axis is (1 + r)/(l — r), where EJE + = r.
If the amplitudes have a phase difference between them, E_/E + = re m ,
then it is easy to show that the ellipse traced out by the E vector has its
axes rotated by an angle (a/2). Figure 7.4 shows the general case of elliptical
polarization and the ellipses traced out by both E and B at a given point in
space.
For r = ± 1 we get back a linearly polarized wave.
208 Classical Electrodynamics
7.3 Superposition of Waves in One Dimension; Group Velocity
In the previous sections planewave solutions to Maxwell's equations
were found and their properties discussed. Only monochromatic waves,
those with a definite frequency and wave number, were treated. In actual
circumstances such idealized solutions do not arise. Even in the most
monochromatic light source or the most sharply tuned radio transmitter
or receiver, one deals with a finite (although perhaps small) spread of
frequencies or wavelengths. This spread may originate in the finite
duration of a pulse, in inherent broadening in the source, or in many
other ways. Since the basic equations are linear, it is in principle an
elementary matter to make the appropriate linear superposition of
solutions with different frequencies. In general, however, there are several
new features which arise.
1. If the medium is dispersive (i.e., the dielectric constant is a function
of the frequency of the fields), the phase velocity is not the same for each
frequency component of the wave. Consequently different components of
the wave travel with different speeds and tend to change phase with respect
to one another. This leads to a change in the shape of a pulse, for example,
as it travels along.
2. In a dispersive medium the velocity of energy flow may differ greatly
from the phase velocity, or may even lack precise meaning.
3. In a dissipative medium, a pulse of radiation will be attenuated as it
travels with or without distortion, depending on whether the dissipative
effects are or are not sensitive functions of frequency.
The essentials of these dispersive and dissipative effects are implicit in
the ideas of Fourier series and integrals (Section 2.9). For simplicity, we
consider scalar waves in only one dimension. The scalar amplitude
u(x, i) can be thought of as one of the components of the electromagnetic
field. The basic solution to the wave equation (7.2) has been exhibited in
(7.6). The relationship between frequency w and wave number k is given
by (7.5) for the electromagnetic field. Either co or k can be viewed as the
independent variable when one considers making a linear superposition.
Initially we will find it most convenient to use k as an independent variable.
To allow for the possibility of dispersion we will consider co as a general
function of k:
co = co(k) (7.25)
Since the dispersive properties cannot depend on whether the wave travels
to the left or to the right, co must be an even function of k, co( —k) =
oj(k). For most wavelengths co is a smoothly varying function of A:. But at
[Sect. 7.3] Plane Electromagnetic Waves 209
certain frequencies there are regions of "anomalous dispersion" where to
varies rapidly over a narrow interval of wavelengths. With the general form
(7.25), our subsequent discussion can apply equally well to electromagnetic
waves, sound waves, de Broglie matter waves, etc. For the present we
assume that k and <o{k) are real, and so exclude dissipative effects.
From the basic solutions (7.6) we can build up a general solution of the
form
u(x, i) = 4= 1*°° A(k)e ikx  it0(k)t dk (7.26)
The factor I/VItt has been inserted to conform with the Fourier integral
notation of (2.50) and (2.51). The amplitude A(k) describes the properties
of the linear superposition of the different waves. It is given by the
transform of the spatial amplitude u(x, t), evaluated at t = 0*:
A(k) = j= f °° u(x, 0)e ikx dx (7.27)
J2TT J co
If u(x, 0) represents a harmonic wave e ik » x for all x, the orthogonality
relation (2.52) shows that A(k) = V2^d(k — k ), corresponding to a
monochromatic traveling wave u(x, i) = e^"^ as required. If,
however, at t = 0, u(x, 0) represents a finite wave train with a length of
order Ax, as shown in Fig. 7.5, then the amplitude A(k) is not a delta
function. Rather, it is a peaked function with a breadth of the order of Ak,
centered around a wave number k which is the dominant wave number in
the modulated wave u(x, 0). If Ax and Ak are defined as the rms deviations
from the average values of x and k [defined in terms of the intensities
\u(x, 0) 2 and i4(fc) 2 ], it is possible to draw the general conclusion:
AxAk>% (7.28)
The reader may readily verify that, for most reasonable pulses or wave
packets which do not cut off too violently, Ax times Ak lies near the lower
limiting value in (7.28). This means that short wave trains with only a
few wavelengths present have a very wide distribution of wave numbers of
monochromatic waves, and conversely that long sinusoidal wave trains
are almost monochromatic. Relation (7.28) applies equally well to
distributions in time and in frequency.
The next question is the behavior of a pulse or finite wave train in time.
* The following discussion slights somewhat the initialvalue problem. For a second
order differential equation we must specify not only u(x, 0) but also du(x, 0)1 dt. This
omission is of no consequence for the rest of the material in this section. It is remedied in
the following section.
210
u(x, 0)
Classical Electrodynamics
A(k)
Fig. 7.5 A harmonic wave train
of finite extent and its Fourier
spectrum in wave number.
The pulse shown at t = in Fig. 7.5 begins to move as time goes on. The
different frequency or wavenumber components in it move at different
phase velocities. Consequently there is a tendency for the original
coherence to be lost and for the pulse to become distorted in shape. At the
very least, we might expect it to propagate with a rather different velocity
from, say, the average phase velocity of its component waves. The general
case of a highly dispersive medium or a very sharp pulse with a great
spread of wave numbers present is difficult to treat. But the propagation
of a pulse which is not too broad in its wavenumber spectrum, or a pulse
in a medium for which the frequency depends weakly on wave number, can
be handled in the following approximate way. The wave at time t is given
by (7.26). If the distribution A(k) is fairly sharply peaked around some
value k , then the frequency oy{k) can be expanded around that value of k:
a)(k) = w + —
dk
(k  k ) +
(7.29)
and the integral performed. Thus
e ilk (d<o/dk)\ (o ]t /*oo
u(x, t) ~ — ,4(fe) e «*(*»/*)Io*]* dk (7 30)
yJllT J oo
From (7.27) and its inverse it is apparent that the integral in (7.30) is just
u{x', 0), where x' = x — (da>ldk)\ t:
u(x, t)~ ulx — t, o)e i[ ' Co(< * to/< ** ) lo £,, o]* / 7 2i)
V dk o /
[Sect. 7.3] Plane Electromagnetic Waves 211
This shows that, apart from an overall phase factor, the pulse travels along
undistorted in shape with a velocity, called the group velocity:
dco
(7.32)
If an energy density is associated with the magnitude of the wave (or its
absolute square), it is clear that in this approximation the transport of
energy occurs with the group velocity, since that is the rate at which the
pulse travels along.
For light waves the relation between co and k is given by
a>(k) = % (7.33)
n(k)
where c is the velocity of light in vacuum, and n(k) is the index of refraction
expressed as a function of k. The phase velocity is
,, = ^ = S (7.34)
k n{k)
and is greater or smaller than c depending on whether n{k) is smaller or
larger than unity. For most optical wavelengths n{k) is greater than unity
in almost all substances. The group velocity (7.32) is
v a = (7.35)
\n(co) + co(dn/dco)]
In this equation it is more convenient to think of n as a function of co than
of k. For normal dispersion (dn/dco) > 0, and also n > 1 ; then the
velocity of energy flow is less than the phase velocity and also less than c.
In regions of anomalous dispersion, however, dn/dco can become large and
negative. Then the group velocity differs greatly from the phase velocity,
often becoming larger than c. * The behavior of group and phase velocities
as a function of frequency in the neighborhood of a region of anomalous
dispersion is shown in Fig. 7.6.
* There is no cause for alarm that our ideas of special relativity are here violated;
group velocity is no longer a meaningful concept. A large value of dn/dco is equivalent to
a rapid variation of eo as a function of k. Consequently the approximations made in
(7.29) ff. are no longer valid. The behavior of the pulse is much more involved.
212
Classical Electrodynamics
n (go)
Fig. 7.6 Index of refraction n(co)
as a function of frequency eo at
a region of anomalous disper
sion ; phase velocity v„ and group
velocity v g as functions of to.
7.4 Illustration of Propagation of a Pulse in a Dispersive Medium
To illustrate the ideas of the previous section and to show the validity
of the concept of group velocity we will now consider a specific model for
the dependence of frequency on wave number and will calculate without
approximations the propagation of a pulse in this model medium. Before
specifying the particular model it is necessary to state the initialvalue
problem in more detail than was done in (7.26) and (7.27). As noted there,
the proper specification of an initial value problem for the wave equation
demands the initial values of both function u(x, 0) and time derivative
du(x, 0)/dt. If we agree to take the real part of (7.26) to obtain u(x, t),
i i r°°
u(x, =  == A (k)e ikx  i(oWt dk + c. c. (7.36)
then it is easy to show that A(k) is given in terms of the initial values by:
_ 1_ I °° ihx, / twx . i du
J2.TT J oo
A(k)
u(x,0) +
(*,0)
dx (7.37)
co(k) dt
We will take a Gaussian modulated oscillation
u(x, 0) = e ~ x2/2L * cos k x (7.38)
as the initial shape of the pulse. For simplicity, we will assume that
— (x, 0) = (7.39)
[Sect. 7.4] Plane Electromagnetic Waves 213
This means that at times immediately before t = the wave consisted of
two pulses, both moving towards the origin, such that at t = they
coalesced into the shape given by (7.38). Clearly at later times we expect
each pulse to reemerge on the other side of the origin. Consequently the
initial distribution (7.38) may be expected to split into two identical
packets, one moving to the left and one to the right. The Fourier amplitude
A(£) for the pulse described by (7.38) and (7.39) is:
A (fy = _4= °° e ikx e x*/2L 2 cos koX dx
JItT J oo
Lr e (L 2 /2)(kk ) 2 _j_ e (i 2 /2)(fc + fc ) 2 j (740)
The symmetry A(—k) = A(k) is a reflection of the presence of two pulses
traveling away from the origin, as will be seen below.
In order to calculate the wave form at later times we must specify
o = a)(k). As a model allowing exact calculation and showing the
essential dispersive effects, we assume
co(k) = v{l + — ) (741)
where v is a constant frequency, and a is a constant length which is a typical
wavelength where dispersive effects become important. Since the pulse
(7.38) is a modulated wave of wave number k = k , the approximate
arguments of the preceding section imply that the two pulses will travel
with the group velocity
v g = ^(k ) = va"k (7.42)
dk
and will be essentially unaltered in shape provided the pulse is not too
narrow in space.
The exact behavior of the wave as a function of time is given by (7.36),
with (7.40) for A(k)\
u ( x t \ = L _ Ref 00 r e (L 2 /2)Vck Q ) 2 + e (£ 2 /2)(fc + k ) 2 y**iv<[l + (aV/2)] rffc
lyJllT J oo
(7.43)
214
Classical Electrodynamics
The integrals can be performed by appropriately completing the squares
in the exponents. The result is
u(x, i) =  Re
exp
(x — va 2 k t) 2
(' + f1
2L 2 1 +
(i + ^J
x exp
jfc a; — iv\ 1 H =i 1 r
+ (h^k Q )
(7.44)
Equation (7.44) represents two pulses traveling in opposite directions.
The peak amplitude of each pulse travels with the group velocity (7.42),
while the modulation envelope remains Gaussian in shape. The width of
the Gaussian is not constant, however, but increases with time. The width
of the envelope is
L(r) =
L 2 +
m
(7.45)
Thus the dispersive effects on the pulse are greater (for a given elapsed
time), the sharper the envelope. The criterion for a small change in shape
is that L > a. Of course, at long times the width of the Gaussian increases
linearly with time
L
(7.46)
but the time of attainment of this asymptotic form depends on the ratio
(L/d). A measure of how rapidly the pulse spreads is provided by a com
parison of L(t) given by (7.45), with v g t = va 2 k t. Figure 7.7 shows two
examples of curves of the position of peak amplitude (v g t) and the positions
v g t ± L(t), which indicate the spread of the pulse, as functions of time. On
the left the pulse is not too narrow compared to the wavelength k ~ x and
so does not spread too rapidly. The pulse on the right, however, is so
narrow initially that it is very rapidly spread out and scarcely represents a
pulse after a short time.
Although the above results have been derived for a special choice (7.38)
of initial pulse shape and dispersion relation (7.41), their implications are
of a more general nature. We have seen in Section 7.3 that the average
velocity of a pulse is the group velocity v g = da>/dk = co'. The spreading
of the pulse can be accounted for by noting that a pulse with an initial
[Sect. 7.4]
Plane Electromagnetic Waves
215
koL<\
Fig. 7.7 Change in shape of a wave packet as it travels along. The broad packet,
containing many wavelengths (k„L > 1), is distorted comparatively little, while the
narrow packet (k L < 1) broadens and diffuses out rapidly.
spatial width A:r must have inherent in it a spread of wave numbers
Afc ~ (1/A» ). This means that the group velocity, when evaluated for
various k values within the pulse, has a spread in it of the order
Ap a
A#„
(7.47)
At a time t this implies a spread in position of the order of Aty. If we
combine the uncertainties in position by taking the square root of the sum
of squares, we obtain the width Asc(0 at time t :
A^^VcA^+g) 2
(7.48)
We note that (7.48) agrees exactly with (7.45) if we put Aa; = L. The
expression (7.48) for Arc(0 shows the general result that, if co" =£ 0, a
narrow pulse spreads rapidly because of its broad spectrum of wave
numbers, and vice versa. All these ideas carry over immediately into wave
mechanics. They form the basis of the Heisenberg uncertainty principle.
In wave mechanics, the frequency is identified with energy divided by
Planck's constant, while wave number is momentum divided by Planck's
constant.
The problem of wave packets in a dissipative, as well as dispersive,
medium is rather complicated. Certain aspects can be discussed analyti
cally, but the analytical expressions are not readily interpreted physically.
Wave packets are attenuated and distorted appreciably as they propagate.
The reader may refer to Stratton, pp. 301309, for a discussion of the
problem, including numerical examples.
216
Classical Electrodynamics
7.5 Reflection and Refraction of Electromagnetic Waves at a Plane
Interface between Dielectrics
The reflection and refraction of light at a plane surface between two
media of different dielectric properties are familiar phenomena. The
various aspects of the phenomena divide themselves into two classes.
(1) Kinematic properties :
(a) Angle of reflection equals angle of incidence.
sin i ri
(b) SnelPs law : — — = — , where i, r are the angles of incidence
sin r n
and refraction, while n, n' are the corresponding indices of re
fraction.
(2) Dynamic properties :
(a) Intensities of reflected and refracted radiation.
(b) Phase changes and polarization.
The kinematic properties follow immediately from the wave nature of
the phenomena and the fact that there are boundary conditions to be
satisfied. But they do not depend on the nature of the waves or the
boundary conditions. On the other hand, the dynamic properties depend
entirely on the specific nature of electromagnetic fields and their boundary
conditions.
The coordinate system and symbols appropriate to the problem are
shown in Fig. 7.8. The media below and above the plane 2 = have
permeabilities and dielectric constants //, e and (x ', e, respectively. A plane
wave with wave vector k and frequency co is incident from medium fi, e.
The refracted and reflected waves have wave vectors k' and k", respectively,
and n is a unit normal directed from medium [i, e into medium [x, e'.
Fig. 7.8 Incident wave k strikes
plane interface between different
media, giving rise to a reflected
wave k" and a refracted wave k'.
[Sect. 7.5] Plane Electromagnetic Waves 217
According to (7.18), the three waves are
INCIDENT
E = E e ik  x  iwt )
BVF — (?  49)
REFRACTED
REFLECTED
e' = E y k "*" ia * )
E" = E V k " x  ta " )
— k" x E" (7.51)
B" = yltu
k
The wave numbers have the magnitudes
CO / —
c
CO
c
ik = k"i = /c = v /Me
c
ik'i = k' =  v^
(7.52)
The existence of boundary conditions at z = 0, which boundary
conditions must be satisfied at all points on the plane ,at all times, implies
that the spatial (and time) variation of all fields must be the same at z = 0.
Consequently, we must have the phase factors all equal at z = 0,
(k • x) 2=0 = (k' . x), =0 = (k" • x) z=0 (7.53)
independent of the nature of the boundary conditions. Equation (7.53)
contains the kinematic aspects of reflection and refraction. We see
immediately that all three wave vectors must lie in a plane. Furthermore,
in the notation of Fig. 7.8,
k sin / = k' sin r = k" sin r' (7.54)
Since k" = k, we find i = r'; the angle of incidence equals the angle of
reflection. SnelFs law is
!"Li = ^' = /^'= (7.55)
sinr k A* e n
The dynamic properties are contained in the boundary conditions —
normal components of D and B are continuous; tangential components of
218
Classical Electrodynamics
Fig. 7.9 Reflection and refraction
with polarization perpendicular to
the plane of incidence.
E and H are continuous. In terms of fields (7.49)(7.51) these boundary
conditions at z = are :
[ e (E + E ")e'E '].n = 0l
[k x E + k" x E "  k' x E '] • n =
(E + E "  E ') x n =
 (k x E + k" x E ")   f (k' x E ')
x n =
(7.56)
In applying these boundary conditions it is convenient to consider two
separate situations, one in which the incident plane wave is linearly
polarized with its polarization vector perpendicular to the plane of
incidence (the plane defined by k and n), and the other in which the
polarization vector is parallel to the plane of incidence. The general case
of arbitrary elliptic polarization can be obtained by appropriate linear
combinations of the two results, following the methods of Section 7.2.
We first consider the electric field perpendicular to the plane of incidence,
as shown in Fig. 7.9. All the electric fields are shown directed away from
the viewer. The orientations of the B vectors are chosen to give a positive
flow of energy in the direction of the wave vectors. Since the electric
fields are all parallel to the surface, the first boundary condition in (7.56)
yields nothing. The third and fourth equations in (7.56) give
^o + Eq" ~ Eq =
 (#o  V) cos iJ — E o cos r =
(7.57)
[Sect. 7.5]
Plane Electromagnetic Waves
219
while the second, using Snell's law, duplicates the third. The relative
amplitudes of the refracted and reflected waves can be found from (7.57).
These are:
E PERPENDICULAR TO PLANE OF INCIDENCE
E n '
1 +
E"
1 
ft' tan r
ju tan /
ft' tan r
1 +
ju tan i
ft' tan r
2 cOs i sin r
ft tan i sin (/ + r)
sin (t — r)
sin (i + r)
(7.58)
The expression on the right in each case is the result appropriate for
[i' = fi,, as is generally true for optical frequencies.
If the electric field is parallel to the plane of incidence, as shown in Fig.
7.10, the boundary conditions involved are normal D, tangential E, and
tangential H [the first, third, and fourth equations in (7.56)]. The
tangential E and H continuous demand that
cos i (E — Eq) — cos r E ' =
1(E + E ")Jle '=0
fX * ft'
(7.59)
Normal D continuous, plus Snell's law, merely duplicates the second of
these equations. The relative amplitudes of refracted and reflected fields
are therefore
Fig. 7.10 Reflection and refrac
tion with polarization parallel to
the plane of incidence.
E'
k'
n>
&B'
MV
tit
E
k /
/ B
i \
B
\
220 Classical Electrodynamics
E PARALLEL TO PLANE OF INCIDENCE
E '
= 2
/*
/U€
sin 1i
2 cos i sin r
we' ^ . sin (i + r) cos (i — r)
^ sin 2r + — sin 2i '
„ „ ~ sin 2/ — sin 2r
tan (i — r)
E . fj, . . tan (i + r)
sin 2r + — sin 2i
Again the results on the right apply for fi = fi.
For normal incidence (/ = 0), both (7.58) and (7.60) reduce to
(7.60)
E n '
In
Eo
it* i
n' + n
E " '
v fx'e
n' — n
E
M+i
n' + n
(7.61)
For the reflected wave the sign convention is that for polarization parallel
to the plane of incidence. This means that if ri > n there is a phase
reversal for the reflected wave.
7.6 Polarization by Reflection and Total Internal Reflection
Two aspects of the dynamical relations on reflection and refraction are
worthy of mention. The first is that for polarization parallel to the plane
of incidence there is an angle of incidence, called Brewster's angle, for
which there is no reflected wave. Putting p = /u for simplicity, we see
from (7.60) that there will be no reflected wave when i + r = ttJ2. From
Snell's law (7.55) we find that this specifies Brewster's angle to be
in — tan x
o
(7.62)
For a typical ratio (n'jn) = 1.5, i B ~ 56°. If a plane wave of mixed
polarization is incident on a plane interface at the Brewster angle, the
reflected radiation is completely plane polarized with polarization vector
perpendicular to the plane of incidence. This behavior can be utilized to
[Sect. 7.6] Plane Electromagnetic Waves 221
produce beams of planepolarized light, but is not as efficient as other
means employing anisotropic properties of some dielectric media. Even
if the unpolarized wave is reflected at angles other than the Brewster angle,
there is a tendency for the reflected wave to be predominantly polarized
perpendicular to the plane of incidence. The success of dark glasses which
selectively transmit only one direction of polarization depends on this fact.
In the domain of radiofrequencies, receiving antennas can be so oriented
as to discriminate against surfacereflected waves (and also waves reflected
from the ionosphere) in favor of the directly transmitted wave.
The second phenomenon is called total internal reflection. The word
internal implies that the incident and reflected waves are in a medium of
larger index of refraction than the refracted wave (n > n'). Snell's law
(7.55) shows that, if n > «', then r > i. Consequently, r = tt\2 when
i = i , where
)
n!
; = sin" 1 (j (7.63)
For waves incident at / = / , the refracted wave is propagated parallel to
the surface. There can be no energy flow across the surface. Hence at
that angle of incidence there must be total reflection. What happens if
/ > i ? To answer this we first note that, for i > /„, sin r > 1 . This means
that r is a complex angle with a purely imaginary cosine.
sin
cos r = i
5«Li)*_i (7.64)
sin iJ
sin i
The meaning of these complex quantities becomes clear when we consider
the propagation factor for the refracted wave :
iW x ik'(x sin r+z cos r) — A'[(sin i/sln i ) 2 — l]^z ik' (sin i/ sin i )x (1 f*S\
This shows that, for / > i , the refracted wave is propagated only parallel
to the surface and is attenuated exponentially beyond the interface. The
attenuation occurs within a very few wavelengths of the boundary, except
for i ^ /„.
Even though fields exist on the other side of the surface it is clear that
there is no energy flow through the surface. Hence total internal reflection
occurs for / > i . The lack of energy flow can be verified by calculating
the timeaveraged normal component of the Poynting's vector just inside
the surface:
S • n = — Re [n • (E' x H'*)] (7.66)
222 Classical Electrodynamics
c
with H' = — (k' x E'), we find
JU CO
S • n = ^£ t Re [(n . k') E ' 2 ] (7.67)
But n • k' = k' cos r is purely imaginary, so that S • n = 0.
The phenomenon of total internal reflection is exploited in many
applications where it is required to transmit light without loss in intensity.
In nuclear physics Lucite or other plastic "light pipes" are used to carry
light emitted from a scintillation crystal because of the passage of an
ionizing particle to a photomultiplier tube, where it is converted into a
useful electric signal. The photomultiplier must often be some distance
away from the scintillation crystal because of space limitations or magnetic
fields which disturb its performance. If the light pipe is large in cross
section compared to a wavelength of the radiation involved, the con
siderations presented here for a plane interface have approximate validity.
When the dielectric medium has crosssectional dimensions of the order
of a wavelength, however, the precise geometry must be taken into account.
Then the propagation is that of a dielectric wave guide (see Section 8.8).
7.7 Waves in a Conducting Medium
If the medium in which waves are propagating is a conductor, there are
characteristic differences in the propagation, when compared with non
conducting media. If the medium is characterized by a conductivity a,
as well as a dielectric constant e and permeability /u, Maxwell's equations
are supplemented by Ohm's law :
J = <rE (7.68)
Hence they take the form :
V./*H = VxE + ^— =0
c dt
V.eE = V xH^^E =
c dt c
(7.69)
In the insulating dielectric we found that the timevarying fields were
transverse, i.e., the field vectors E and H were perpendicular to the
direction in which the spatial variation occurred. In the limit of zero
frequency we know from our study of electro and magnetostatics that the
static fields in a dielectric are longitudinal, in the sense that the fields are
derivable from scalar potentials and so point in the direction of the spatial
variation.
[Sect. 7.7]
Plane Electromagnetic Waves
223
If the conductivity is not zero, modifications arise. For simplicity,
consider fields which vary in only one spatial variable, . We decompose
the fields into longitudinal and transverse parts :
E(£, = E long (, + E tr (,
H(£, = H long (£, + H tr (£, j
(7.70)
Then, because of the properties of curl operation, we find that the trans
verse parts of E and H satisfy the two curl equations in (7.69), leading to
transverse waves (see below), while the longitudinal parts satisfy the
equations :
ilong __ q
31
long
dt
dE
long
dS
=
/ 3 477(7
\dt e
Eiong =
(7.71)
The first pair of equations shows that the only longitudinal magnetic field
possible is a static uniform field. This is the same situation as in an
insulator. But the second pair in (7.71) shows that the longitudinal
electric field is uniform in space, while having the time variation :
£iong(l, = E e^ t/€
(7.72)
Consequently, no static longitudinal fields can exist in a conducting
medium in the absence of an applied current density. For good conductors
like copper, a ~ 10 17 sec 1 so that disturbances are damped out in an
extremely short time.
We now consider the transverse fields in the conducting medium.
Assuming that the fields vary as exp (/k • x — icot), the first curl equation
of (7.69) yields:
H = — (k x E) (7.73)
flOO
while the second gives
.,, __. , . CO _, 477(7 _ _
i(k x H) + ic — E E =
c c
(7.74)
Elimination of either H or E from this last equation with (7.73) yields
; 2 ( co 2
+ 4777
jua>a\
=
This means that the propagation vector k is complex :
fc 2 = [it
\ coe I
(7.75)
(7.76)
224
Classical Electrodynamics
The first term corresponds to the displacementcurrent and the second to
the conductioncurrent contribution. In taking the square root to find k
the branch is chosen to give the familiar results when a = 0. Then one
finds, assuming that a is real,
where
k = a + ifl
a / — co
1 +
(£)'
± 1
(7.77)
(477(7 \
< 1 I we find approximately
k = a + iff — yjfxe \ i
Itt ///,
(7.78)
correct to first order in (a/coe). In this limit Re A: > Im k and the attenua
tion of the wave (Im k) is independent of frequency, aside from the possible
frequency variation of the conductivity. For a good conductor I > 1 I,
on the other hand, a and /? are approximately equal: v (xi€
k ^ (I + /) ^ 27ra ^ g
(7.79)
where only the lowestorder terms in (toe/a) have been kept.
The waves propagating as exp (/k • x — icot) are damped, transverse
waves. The fields can be written as
E = E n e~ pn  x e ian  x  i(Ut ]
„ . , (7.80)
H = H e (in *e ma  x  l( " t J
where n is a unit vector in the direction of k. The divergence equation for
E shows that E • n = 0, while the relation between H and E (7.73) gives
H = — (a + i/8)n x E
[XOi
(7.81)
This shows that H and E are out of phase in a conductor. Defining the
magnitude and phase of k :
\k\ = Va 2 + P = V^e
1 + (^
2
CO€ / 
"1
<f> = tan 1  =  tan 1
a
/ 47T(t \
\ iO€ J
(7.82)
[Sect. 7.8] Plane Electromagnetic Waves 225
equation (7.81) can be written in the form:
\ toe/
Ho=./ 1
1 +
e*n x E (7.83)
The interpretation of (7.83) is that H lags E in time by the phase angle <f>
and has a relative amplitude :
\M = fih + l^fT' (7.84)
E  V [x L \ coe I J
In very good conductors we see that the magnetic field is very large com
pared to the electric field and lags in phase by almost 45°. The field energy
is almost entirely magnetic in nature.
The waves given by (7.80) show an exponential damping with distance.
This means that an electromagnetic wave entering a conductor is damped
to 1/e = 0.369 of its initial amplitude in a distance:
6 =  ~ . C (7.85)
the last form being the approximation for good conductors. The distance
6 is called the skin depth or the penetration depth. * For a conductor like
copper, d ^ 0.85 cm for frequencies of 60 cps, and d ~ 0.71 x 10~ 3 cm
for 100 Mc/sec. This rapid attenuation of waves means that in high
frequency circuits current flows only on the surface of the conductors.
One simple consequence is that the highfrequency inductance of circuit
elements is somewhat smaller than the lowfrequency inductance because
of the expulsion of flux from the interior of the conductors.
The problem of reflection and refraction at an interface between con
ducting media is rather complicated and will not be treated here. The
interested reader may refer to Stratton, pp. 500 ff., for a discussion of this
point. See, however, Section 8.1 for a treatment of fields at the interface
between a dielectric and a good conductor.
7.8 Simple Model for Conductivity
The simplest model of conduction, due originally to Drude (1900), is
that in a metal there are a certain number n of electrons per unit volume
free to move under the action of applied electric fields, but subject to
* For reference, the skin depth (7.85) appears in mks units as d = {Ijficoa) 1 ^.
226 Classical Electrodynamics
damping force due to collisions. Thus the equation of motion of such an
electron is
d\
m — + mgy = eE(x, t) (7.86)
at
where g is the damping constant.* For rapidly oscillating fields the
displacement of the electron is small compared to a wavelength so that
approximately
m — + mgy = eE Q e~ i<at (7.87)
dt
where E is the electric field at the average position of the electron. The
steadystate solution for the velocity of the electron is :
v = — — E e imt (7.88)
m(g — ico)
so that the conductivity is given by
a = "° g2 (7.89)
m{g  ico)
Assuming one free electron per atom, a metal such as copper (« ^ 8 x
10 22 electrons/cm 3 , c^5 x 10 17 sec 1 ) has an empirical damping constant
g^3x 10 13 sec 1 . This shows that for frequencies of the order of, or
smaller than, microwave frequencies (~10 10 sec 1 ) metallic conductivities
are essentially real (i.e., current in phase with the field) and independent of
frequency. At higher frequencies (in the infrared and beyond), however,
the conductivity is complex and depends markedly on frequency in a
manner qualitatively described by the simple result (7.89).
7.9 Transverse Waves in a Tenuous Plasma
In certain situations, such as the ionosphere or a tenuous plasma, the
damping of the motion of the free electrons due to collisions becomes
negligible. Then the "conductivity" becomes purely imaginary:
2
^plasma ^ * — (7.90)
mco
* The damping constant^ is some sort of average rate of collisions involving appreci
able momentum transfer. Collisions occur between electrons and lattice vibrations,
lattice imperfections, and impurities. The proper calculation of g involves quantum
mechanical considerations, including the effects of the Pauli exclusion principle. See
A. H. Wilson, Theory of Metals, 2nd ed., Cambridge University Press (1953).
[Sect. 7.9] Plane Electromagnetic Waves 227
Quotation marks are placed on "conductivity" because there is no resistive
loss of energy if the current and electric field are out of phase. The
propagation of transverse electromagnetic waves in a tenuous plasma is
governed by equation (7.76) of Section 7.7, with cr plasma (7.90) inserted for
a:*
k sV^) (7  91 >
where
coJS _ w (7 92)
m
is called the plasma frequency . Since the wave number can be written as
k = nco/c, where n is the index of refraction, we see that the index of
refraction of a plasma is given by
For highfrequency radiation (eo > co p ) the index of refraction is real and
the waves propagate freely. For frequencies lower than the plasma
frequency w v , n is purely imaginary. Consequently such electromagnetic
waves incident on a plasma will be reflected from the surface. Within the
plasma the fields will fall off exponentially with distance from the surface.
The penetration depth plMma is given by
<5plasn.a = . ~ — (7.94)
V«„ 2 — (O 2 <»p
the last value being valid for a> < co p . On the laboratory scale, plasma
densities are in the range n ~ 10 12 10 16 electrons/cm 3 . This means m v ~
6 x 10 10 6 x 10 12 sec 1 , so that typical penetration depths are of the order
of 0.5 cm5 x 10~ 3 cm for static or lowfrequency fields. The expulsion
of fields from within a plasma is a wellknown effect in controlled thermo
nuclear processes and is exploited in attempts at confinement of hot
plasmas (see Sections 10.5 and 10.6).
The simple result (7.93) for the index of refraction of a plasma is
modified by the presence of an external static magnetic induction. This
circumstance arises not only in the laboratory, but also in the ionosphere,
where the earth's dipole field provides the external magnetic induction.
To illustrate the influence of the external field we consider the simple
* Sometimes this equation is solved for co 2 as a function of k :
to 2 ~ w/ + c 2 k 2
Then it is called a dispersion relation for co = a)(k).
228 Classical Electrodynamics
problem of a tenuous electronic plasma of uniform density with a strong,
static, uniform, magnetic induction B and transverse waves propagating
parallel to the direction of B . If the amplitude of the electronic motion is
small and collisions are neglected, the equation of motion is approximately :
m — ~ eEe imt + e  x B (7.95)
dt c
where the influence of the B field of the transverse wave has been neglected
compared to the static induction B . It is convenient to consider the
transverse waves as circularly polarized. Then
E = £( Cl ± /eg) (7.96)
while B is in the direction of c 3 . Since we are looking for a steadystate
solution, we will assume that the velocity of the electron is of the form :
v(t) = v(e 1 ±ie 2 )e ilot (7.97)
Then from (7.95), using (7.96), we find immediately
v = E (7.98)
m(co ± a) B )
where co B is the frequency of precession of a charged particle in a magnetic
field,
o> B = ^2. (7.99)
mc
Result (7.98) can be understood by noting that, in a coordinate system
precessing with frequency a> B , the electron is driven by a rotating electric
field of effective frequency co ± oo B , depending on the sign of the circular
polarization.
The current density in the plasma due to electronic motion is
2
J = en v = ^ E (7.100)
m(co ± co B )
When this current density is added to the displacement current, Maxwell's
generalization of Ampere's law becomes :
V x H= i
c
1 T"^ E (7.101)
2
p
oo(co ± co B ).
The factor in square brackets can be interpreted as the dielectric constant
or square of the index of refraction :
n ± 2 = 1 ^! (7.102)
oo(co ± co B )
[Sect. 7.9] Plane Electromagnetic Waves 229
This is the extension of (7.93) to include a static magnetic induction. It is
not completely general, since it applies only to waves propagating along
the static field direction. But even in this simple example we see the
essential characteristic that waves of righthanded and lefthanded circular
polarizations propagate differently. The ionosphere is birefringent. For
propagation in directions other than parallel to the static field B it is
straightforward to show that, if terms of the order of co B 2 are neglected
compared to co 2 and axo B , the index of refraction is still given by (7.102).
But the precession frequency (7.99) is now to be interpreted as that due to
only the component of B parallel to the direction of propagation. This
means that co B in (7.102) is a function of angle — the medium is not only
birefringent, but also anisotropic.
For the ionosphere a typical maximum density of free electrons is
n ~ 10M0 6 electrons/cm 3 , corresponding to a plasma frequency of the
order of (o p ~ 6 x 10 6 6 x 10 7 sec 1 . If we take a value of 0.3 gauss as
representative of the earth's magnetic field, the precession frequency is
co B ~6x 10 6 sec 1 .
Figure 7. 1 1 shows n ± 2 as a function of frequency for two values of the
ratio of (coJa) B ). In both examples there are wide intervals of frequency
where one of n + 2 or n_ 2 is positive while the other is negative. At such
frequencies one state of circular polarization cannot propagate in the
plasma. Consequently a wave of that polarization incident on the plasma
will be totally reflected. The other state of polarization will be partially
transmitted. Thus, when a linearly polarized wave is incident on a plasma,
the reflected wave will be elliptically polarized, with its major axis generally
rotated away from the direction of the polarization of the incident
wave.
The behavior of radio waves reflected from the ionosphere is explicable
in terms of these ideas, but the presence of several layers of plasma with
densities and relative positions varying with height and time makes the
problem considerably more complicated than our simple example. The
electron densities at various heights can be inferred by studying the
reflection of pulses of radiation transmitted vertically upwards. The
number of free electrons per unit volume increases slowly with height in a
given layer of the ionosphere, as shown in Fig. 7.12, reaches a maximum,
and then falls abruptly with further increase in height. A pulse of a given
frequency (o x enters the layer without reflection because of the slow change
in « . When the density n is large enough, however, co p (h^) ~ co x . Then
the indices of refraction (7.102) vanish and the pulse is reflected. The
actual density n where the reflection occurs is given by the roots of the
righthand side of (7.102). By observing the time interval between the
initial transmission and reception of the reflected signal the height h x
230
Classical Electrodynamics
= 2.0
,1
— <N
1
M
f ' !
1^ = 0.5
B
■,^t~ \
i
1 CO
/ "J
In 2
2
3
4
Fig. 7.11 Indices of refraction as a function of frequency for model of the ionosphere
(tenuous electronic plasma in a static, uniform magnetic induction). n±((o) apply to
right and left circularly polarized waves propagating parallel to the magnetic field.
o) B is the gyration frequency; eo„ is the plasma frequency. The two sets of curves
correspond to (oJa> B = 2.0, 0.5.
corresponding to that density can be found. By varying the frequency coj
and studying the change in time intervals the electron density as a function
of height can be determined. If the frequency w x is too high, the index of
refraction does not vanish and very little reflection occurs. The frequency
above which reflections disappear determines the maximum electron
density in a given layer.
m
no(h\)
Fig. 7.12 Electron density as a
function of height in a layer of the
ionosphere (schematic).
[Probs. 7] Plane Electromagnetic Waves 231
REFERENCES AND SUGGESTED READING
The whole subject of optics as an electromagnetic phenomenon is treated authorita
tively by
Born and Wolf.
Their first chapter covers plane waves, polarization, and reflection and refraction, among
other topics. A very complete discussion of plane waves incident on boundaries of
dielectrics and conductors is given by
Stratton, Chapter IX.
Another good treatment of electromagnetic waves in both isotropic and anisotropic
media is that of
Landau and Lifshitz, Electrodynamics of Continuous Media, Chapters X and XI.
A more elementary, but clear and thorough, approach to plane waves and their properties
appears in
Adler, Chu, and Fano, Chapters 7 and 8.
The propagation of waves in dispersive media is discussed in detail in the book by
Brillouin.
The distortion and attenuation of pulses in dissipative materials are covered by
Stratton, pp. 301309.
PROBLEMS
7.1 An approximately monochromatic plane wave packet in one dimension has
the instantaneous form, u(x, 0) = f(x)e ik o x , with /(*) the modulation
envelope. For each of the forms /(*) below, calculate the wavenumber
spectrum \A(k)\ 2 of the packet, sketch \u(x, 0) 2 and /4(&) 2 , evaluate explicitly
the rms deviations from the means, Ax and Ak (defined in terms of the
intensities \u(x, 0) 2 and \A(k)f), and test inequality (7.28).
(a) f(x) = Ne*\ X U 2
(b) f(x) = Ne" 2 * 2 I*
(N(l  a a;) for a a; < 1
(c) f(x) = ,
7 (0 for a \x\ > 1
(d)f(x)
K
N for a; < a
for 1*1 > a
7.2 A plane wave is incident on a layered interface as shown in the figure (p. 232).
The indices of refraction of the three nonpermeable media are n x , n 2 , « 3 .
The thickness of the intermediate layer is d.
(a) Calculate the transmission and reflection coefficients (ratios of
transmitted and reflected Poynting's flux to the incident flux), and sketch
their behavior as a function of frequency for n t = 1, n 2 = 2, n^ = 3 ; n x = 3,
n 2 = 2, » 3 = 1 ; and n x = 2, » 2 = 4, n 3 = 1 .
232
Classical Electrodynamics
Bi
(Jb) The medium n x is part of an optical system (e.g., a lens); medium n 3
is air (w 3 = 1). It is desired to put an optical coating (medium n 2 ) on the
surface so that there is no reflected wave for a frequency <o . What thickness
d and index of refraction /i 2 are necessary?
7.3 Two plane semiinfinite slabs of the same uniform, isotropic, nonpermeable,
lossless dielectric with index of refraction n are parallel and separated by an
air gap (n = 1) of width d. A plane electromagnetic wave of" frequency to
is incident on the gap from one of the slabs with angle of incidence i. For
linear polarization both parallel to and perpendicular to the plane of
incidence,
(a) calculate the ratio of power transmitted into the second slab to the
incident power and the ratio of reflected to incident power;
(b) for i greater than the critical angle for total internal reflection, sketch
the ratio of transmitted power to incident power as a function of d measured
in units of wavelength in the gap.
7.4 A plane polarized electromagnetic wave of frequency to in free space is
incident normally on the flat surface of a nonpermeable medium of
conductivity a and dielectric constant e.
(a) Calculate the amplitude and phase of the reflected wave relative to the
incident wave for arbitrary a and e.
(b) Discuss the limiting cases of a very poor and a very good conductor,
and show that for a good conductor the reflection coefficient (ratio of
reflected to incident intensity) is approximately
R
c
where 8 is the skin depth.
7.5 A plane polarized electromagnetic wave E = E i e ik '* i<ot is incident normally
on a flat uniform sheet of an excellent conductor (<r > to) having a thickness t .
Assuming that in space and in the conducting sheet /* = e = 1, discuss the
reflection and transmission of the incident wave.
(a) Show that the amplitudes of the reflected and transmitted waves,
correct to the first order in (to\afA, are:
Er (1  fl(l  e™)
E t
Ei
(1  e™) + 0(1 + 3e 2 *)
4/Se" A
(1  e~ u ) + 0(1 + 3e~ 2A )
[Probs. 7] Plane Electromagnetic Waves 233
where
A = (1  i)t/d
and d = c/ V27rco(r is the penetration depth.
(ft) Verify that for zero thickness and infinite thickness you obtain the
proper limiting results.
(c) Show that, except for sheets of very small thickness, the transmission
coefficient is
_ 32(Re PfeW
1  2e 2 ^ cos (It IS) + *«/«
Sketch log 7* as a function of (t/d), assuming Re £ = lO" 2 .
Define "very small thickness."
7.6 Plane waves propagate in a homogeneous, nonpermeable, but anisotropic
dielectric. The dielectric is characterized by a tensor e w , but if coordinate
axes are chosen as the principal axes the components of displacement along
these axes are related to the electricfield components by D t = e t E t
(/ = 1, 2, 3), where e f are the eigenvalues of the matrix e„.
(a) Show that plane waves with frequency a> and wave vector k must
satisfy
to 2
kx(kxE)+ 1 D=0
c £
(b) Show that for a given wave vector k = kn there are two distinct
modes of propagation with different phase velocities v = co/k which satisfy
the Fresnel equation,
2
2
=
_ „ .2
where v t = c/V €i is called a principal velocity, and /i, is the component of
n along the /th principal axis.
(c) Show that D a • D 6 = 0, where D a , D 6 are the displacements associated
with the two modes of propagation.
7.7 A homogeneous, isotropic, nonpermeable dielectric is characterized by an
index of refraction n(co) which is in general complex in order to describe
absorptive processes.
(a) Show that the general solution for plane waves in one dimension can
be written
1 f 00
u(x, t) = —j= dco e i<ot [A(co)e i ^ a,lc '> n ^ x + B((o)e~ *(<°kX<»)x\
where u(x, t) is a component of E or B.
(b) If u(x, t) is real, show that n(—m) = n*((o).
(c) Show that, if «(0, t) and 3w(0, t)\dx are the boundary values of u
and its derivative at * = 0, the coefficients A(co) and B(a>) are
A(co)\ 1 1 f 00 . T ic du 1
234 Classical Electrodynamics
7.8 A very long planewave train of frequency co with a sharp front edge is
incident normally from vacuum on a semiinfinite dielectric described by
an index of refraction n((o) and occupying the halfspace x > 0. Just
outside the dielectric (at x = 0) the incident electric field is
£,,(0, i) — d(t)e~ et sin co t
where 0(0 is the step function (0(0 = for t < 0, 0(0 = 1 for t > 0). The
exponential decay constant e is a positive infinitesimal.
(a) Using the results of Section 7.5 determine the transmitted field
E '(x, at any point in the dielectric as an integral over real frequencies.
(b) Prove that a sufficient condition for causality (that no signal propagate
faster than the speed of light in vacuum) in this problem is that the index of
refraction as a function of complex co be an analytic function, regular in the
upper half co plane with nonvanishing imaginary part there, and approaching
unity for \co\ »■ <x>.
(c) Generalize the argument of (b) to apply to any incident wave train.
7.9 (a) Show that, if the index of refraction n{co) is analytic in the upper half
complex co plane and approaches unity for large \co\, its real and imaginary
parts are related for real frequencies by the dispersion relation,
■" Jo
Re n(a>) = 1 +  P\ —z — 5 Im "(<>>') doi '
where P stands for Cauchy principal value. Write the other dispersion
relation, expressing the imaginary part as an integral over the real.
(b) Show by direct calculation with the dispersion relation that in a
frequency range where resonant absorption occurs there is necessarily
anomalous dispersion.
(c) The elementary classical model for an index of refraction is based on a
collection of damped electronic oscillators and gives an index of refraction,
, , 2^e 2 y f k
n{co) ~ 1 H > — t 3 :
' m Zi cojf — co* — iv k co
k
where co k is the resonant frequency of the fcth type of oscillator, v k its damping
constant, and/ fc the number of such oscillators per atom. Verify that this
index of refraction has the appropriate properties to satisfy the dispersion
relation of (a).
8
Wave Guides
and Resonant Cavities*
Electromagnetic fields in the presence of metallic boundaries form
a practical aspect of the subject of considerable importance. At high fre
quencies where the wavelengths are of the order of meters or less the only
practical way of generating and transmitting electromagnetic radiation
involves metallic structures with dimensions comparable to the wave
lengths involved. In this chapter we consider first the fields in the neigh
borhood of a conductor and discuss their penetration into the surface and
the accompanying resistive losses. Then the problems of waves guided in
hollow metal pipes and of resonant cavities are treated from a fairly
general viewpoint, with specific illustrations included along the way.
Finally, dielectric wave guides are briefly described as an alternative
method of transmission.
* In this chapter certain formulas, denoted by an asterisk on the equation number, are
written so that they can be read as formulas in mks units provided the first factor in
square brackets is omitted. For example, (8.12) is
dPloaa _ T 1 l/wod .
H„
The corresponding equation in mks form is
dPiosa _ /Mid
~da 4~
where all symbols are to be interpreted as mks symbols, perhaps with entirely different
magnitudes and dimensions from those of the corresponding Gaussian symbols.
If an asterisk appears and there is no square bracket, the formula can be interpreted
equally in Gaussian or mks symbols.
General rules for conversion of any equation into its corresponding mks form are
given in Table 3 of the Appendix.
235
236 Classical Electrodynamics
8.1 Fields at the Surface of and within a Conductor
As was mentioned at the end of Section 7.7, theproblem of reflection
and refraction of waves at an interface of two conducting media is some
what complicated. The most important and useful features of the
phenomenon can, however, be obtained with an approximate treatment
valid if one medium is a good conductor. Furthermore, the method, within
its range of validity, is applicable to situations more general than plane
waves incident.
First consider a surface with unit normal n directed outward from a
perfect conductor on one side into a nonconducting medium on the other
side. Then, just as in the static case, there is no electric field inside the
conductors. The charges inside a perfect conductor are assumed to be so
mobile that they move instantly in response to changes in the fields, no
matter how rapid, and always produce the correct surfacecharge density S
(capital 2 is used to avoid confusion with the conductivity or) :
n • D = [4t7]2 (8.1)*
in order to give zero electric field inside the perfect conductor. Similarly,
for timevarying magnetic fields, the surface charges move in response to
the tangential magnetic field to produce always the correct surface current
K:
n x H =
L C J
K (8.2)"
in order to have zero magnetic field inside the perfect conductor. The
other two boundary conditions are on normal B and tangential E :
n • (B — B c ) = 0 (8.3)"
n x (E  E c ) = 0)
where the subscript c refers to the conductor. From these boundary
conditions we see that just outside the surface of a perfect conductor only
normal E and tangential H fields can exist, and that the fields drop abruptly
to zero inside the perfect conductor. This behavior is indicated schemati
cally in Fig. 8.1.
For a good, but not perfect, conductor the fields in the neighborhood
of its surface must behave approximately the same as for a perfect con
ductor. In Section 7.7 we have seen that inside a conductor the fields are
attenuated exponentially in a characteristic length d, called the skin depth.
For good conductors and moderate frequencies, d is a small fraction of
a centimeter. Consequently, boundary conditions (8.1) and (8.2) are
[Sect. 8.1]
Wave Guides and Resonant Cavities
237
E
H
{=0 i—.
(a) (b)
Fig. 8.1 Fields near the surface of a perfect conductor.
approximately true for a good conductor, aside from a thin transitional layer
at the surface.
If we wish to examine that thin transitional region, however, care must
be taken. First of all, Ohm's law (7.68) shows that with a finite conduct
ivity there cannot actually be a surface layer of current, as implied in (8.2).
Instead, the boundary condition on the magnetic field is
n x (H  H c ) =
(8.4)*
To explore the changes produced by a finite, rather than an infinite,
conductivity we employ a successive approximation scheme. First we
assume that just outside the conductor there exists only a normal electric
field E ± and a tangential magnetic field H„, as for a perfect conductor.
The values of these fields are assumed to have been obtained from the
solution of an appropriate boundaryvalue problem. Then we use the
boundary conditions and Maxwell's equations in the conductor to find the
fields within the transition layer and small corrections to the fields outside.
In solving Maxwell's equations within the conductor we make use of the
fact that the spatial variation of the fields normal to the surface is much
more rapid than the variations parallel to the surface. This means that
we can safely neglect all derivatives with respect to coordinates parallel
to the surface compared to the normal derivative.
If there exists a tangential H ( outside the surface, boundary condition
(8.4) implies the same H ( inside the surface. With the neglect of the dis
placement current in the conductor, the curl equations in (7.69) become
E c ~ —V x H c
47TO 
IC
H c =— V x E c
flOi
(8.5)
238 Classical Electrodynamics
where a harmonic variation e~ imt has been assumed. If n is the unit normal
outward from the conductor and £ is the normal coordinate inward into
the conductor, then the gradient operator can be written
neglecting the other derivatives when operating on the fields within the
Conductor. With this approximation (8.5) become:
E c ~ n x — 2
Attg d
„ . ic w 3E C
[JLOi d£
(8.6)
These can be combined to yield
^ 2 (nxH c ) + (nxH c )~0 (8.7)
— CXIW + 
and
n • H c ~ (8.8)
where 6 is the skin depth defined by (7.85). The second equation shows
that inside the conductor H is parallel to the surface, consistent with our
boundary conditions. The solution for H c is :
H c = H M «T*"e«" (8.9)
where H M is the tangential magnetic field outside the surface. From (8.6)
the electric field in the conductor is approximately:
E c ~ /^L (1  i)(n x H M >^ e^ l& (8.10)
These solutions for H and E inside the conductor exhibit the properties
discussed in Section 7.7 : (a) rapid exponential decay, (b) phase difference,
(c) magnetic field much larger than the electric field. Furthermore, they
show that, for a good conductor, the fields in the conductor are parallel
to the surface* and propagate normal to it, with magnitudes which depend
only on the tangential magnetic field H„ which exists just outside the
surface.
* From the continuity of the tangential component of H and the equation connecting
E to V x H on either side of the surface, one can show that there exists in the conductor
a small normal component of electric field, E e n~ (/coe/47r<r)£' i , but this is of the next
order in small quantities compared with (8.10).
[Sect. 8.1]
Wave Guides and Resonant Cavities
239
From the boundary condition on tangential E (8.3) we find that just
outside the surface there exists a small tangential electric field given by
(8.10), evaluated at  = 0:
Ei,
(1  i)(n x H„)
(8.11)
In this approximation there is also a small normal component of B just
outside the surface. This can be obtained from Faraday's law of induction
and gives B ± of the same order of magnitude as E N . The amplitudes of
the fields both inside and outside the conductor are indicated schematically
in Fig. 8.2.
The existence of a small tangential component of E outside the surface,
in addition to the normal E and tangential H, means that there is a power
flow into the conductor. The timeaverage power absorbed per unit area
is
dPu
da
=  ° Re [n • E x H*] = \f\ ^ H„ 2 (8.12)<
8tt UttJ 4
This result can be given a simple interpretation as ohmic losses in the body
E,
H
Fig. 8.2 Fields near the surface of a good, but not perfect, conductor.
240 Classical Electrodynamics
of the conductor. According to Ohm's law, there exists a current density
J near the surface of the conductor:
J = oE e = /^ (1  0(n x H M >^*>/* (8.13)
The timeaverage rate of dissipation of energy per unit volume in ohmic
losses is £J • E* = (l/2<r)  J 2 , so that the total rate of energy dissipation
in the conductor for the volume lying beneath an area element AA is
± AA f°°^J . J* = AA^ HJ 2 f V«" d£ = AA^ IHJ 2
2a Jo 877 Jo 16tt
This is the same rate of energy dissipation as given by the Poynting's
vector result (8.12).
The current density J is confined to such a small thickness just below
the surface of the conductor that it is equivalent to an effective surface
current K eff :
i
Keff = J d£ =
c
Att
n x H„ (8.14)'
Comparison with (8.2) shows that a good conductor behaves effectively
like a perfect conductor, with the idealized surface current replaced by an
equivalent surface current which is actually distributed throughout a very
small, but finite, thickness at the surface. The power loss can be written in
terms of the effective surface current :
dPioss 1 ,^ r .2
da 2ad
Keftr (8.15)*
This shows that 1 /ad plays the role of a surface resistance of the con
ductor. Equation (8.15), with K eff given by (8.14), or (8.12) will allow us
to calculate approximately the resistive losses for practical cavities, trans
mission lines, and wave guides, provided we have solved for the fields in
the idealized problem of infinite conductivity.
8.2 Cylindrical Cavities and Wave Guides
A practical situation of great importance is the propagation or excitation
of electromagnetic waves in hollow metallic cylinders. If the cylinder has
end surfaces, it is called a cavity; otherwise, a wave guide. In our
discussion of this problem the boundary surfaces will be assumed to be
perfect conductors. The losses occurring in practice can be accounted for
Wave Guides and Resonant Cavities
241
[Sect. 8.2]
adequately by the methods of Section 8.1. A cylindrical surface S of
general crosssectional contour is shown in Fig. 8.3. For simplicity, the
crosssectional size and shape are assumed constant along the cylinder axis.
With a sinusoidal time dependence e~ i(0t for the fields inside the cylinder,
Maxwell's equations take the form:
V x E= iB
VB =
VxB=i>eE VE =
(8.16)
where it is assumed that the cylinder is filled with a uniform nondissipative
medium having dielectric constant e and permeability fx. If follows that
both E and B satisfy
("■"30
(8.17)
Because of the cylindrical geometry it is useful to single out the spatial
variation of the fields in the z direction and to assume
E(z, y, z, t)
B(x, y, z, t)
E(x, y)e ±ilez ' i<ot
B(x, y) e ±ikz  i(0t
(8.18)
Appropriate linear combinations can be formed to give traveling or
standing waves in the z direction. The wave number k is, at present, an
unknown parameter which may be real or complex. With this assumed z
dependence of the fields the wave equation reduces to the twodimensional
form:
[*.' + ("?*)]0
where V, 2 is the transverse part of the Laplacian operator:
V, 2 = V 2  —
* dz 2
(8.19)
(8.20)
Fig. 8.3 Hollow, cylindrical wave guide of arbitrary crosssectional shape.
242
Classical Electrodynamics
It is also useful to separate the fields into components parallel to and
transverse to the z axis :
t , E = E, + E, (8.21)
where the parallel field is
E z = (e 3 • E)e 3 (8.22)
and the transverse field is
E t = (e 3 x E) x e 3
(8.23)
and e 3 is a unit vector in the z direction. Similar definitions hold for the
magneticflux density B. Manipulation of the curl equations in (8.16) and
use of the explicit z dependence (8.18) lead to the determination of the
transverse fields in terms of the axial components :
B, =
E, =
("7*)
1
+ i>€  e 3 x V t E z
c
Wf)
(8.24)
These relations show that it is sufficient to determine E z and B, as the
appropriate solutions of the twodimensional wave equation (8.19). The
other components can then be calculated from (8.24).
The boundary values on the surface of the cylinder will be taken as those
for a perfect conductor:
n • B = 0, nxE =
(8.25)
where n is a unit normal at the surface. Since Maxwell's equations and
the boundary conditions are internally consistent, it is sufficient to note
that the vanishing of tangential E at the surface requires
Ez \s = (8.26)
For the normal components of B, using the expression for B t (8.24), we
find that n • B = implies
dB„
dn
=
(8.27)
where d/dn is the normal derivative at a point on the surface.
The twodimensional wave equations (8.19) for E z and B z , together with
the boundary conditions on E z and B z at the surface of the cylinder,
specify eigenvalue problems of the usual sort. For a given frequency co,
only certain values of the axial wave number k will be consistent with
[Sect. 8.2] Wave Guides and Resonant Cavities 243
the differential equation and the boundary conditions (typical waveguide
situation); or, for a given k, only certain frequencies m will be allowed
(typical resonantcavity situation). Because the boundary conditions on
E z and B z are different, they cannot generally be satisfied simultaneously.
Consequently the fields divide themselves into two distinct categories:
TRANSVERSE MAGNETIC (TM)
B z = everywhere
The boundary condition is
E z \s =
TRANSVERSE ELECTRIC (TE)
E z = everywhere
The boundary condition is
an
=
dB,
dn
s
The designations "Electric (or E) Waves" and "Magnetic (or H) Waves"
are sometimes used instead of Transverse Magnetic and Transverse
Electric, respectively, corresponding to specification of the axial com
ponent of the field. In addition to these two types of fields there is a
degenerate mode, called the Transverse Electromagnetic (TEM) mode, in
which both E z and B z vanish. From (8.24) we see that, in order to have
nonvanishing transverse components when both E z and B z vanish, the
axial wave number must satisfy the condition:
k = V^ (8.28)
Thus TEM waves travel as if they were in an infinite medium without
boundary surfaces. From the twodimensional wave equation (8.19) we
now find
V t 2 ( Etem \ = (8.29)
showing that each component of the transverse fields satisfies Laplace's
equation of electrostatics in two dimensions. It is easy to show (a) that
both E TEM and B TEM are derivable from scalar potentials satisfying
Laplace's equation and (b) that B TEM is everywhere perpendicular to E TEM .
In fact, from Faraday's law of induction we find
B TEM = — f(e3*E TEM ) (830)
ICO oz
244 Classical Electrodynamics
With zdependence e iVft€(OZlc , we have
B T EM = V" 6 e 3 X E TEM (8.31)*
which is just the relation for plane waves in an infinite medium.
An immediate consequence of (8.29) is that the TEM mode cannot
exist inside a single hollow, cylindrical conductor of infinite conductivity.
The surface is an equipotential; hence the electric field vanishes inside.
It is necessary to have two or more cylindrical surfaces in order to support
the TEM mode. The familiar coaxial cable and the parallelwire trans
mission line are structures for which this is the dominant mode. (See
Problems 8.1 and 8.2.)
8.3 Wave Guides
We now consider the propagation of electromagnetic waves along a
hollow wave guide of uniform cross section. With the zdependence e ikz ,
the transverse components of the fields for the two types of waves are
related, according to (8.24), as follows:
TM waves: b* = ^ e 3 x E,
ck
TE waves: E(=e 3 xB (
ck
(8.32)
The transverse fields are in turn determined by the longitudinal fields:
TM waves: Et= l ^V t y>
., (8.33)
TE waves: B, = — V t y>
y 2
where ip is E z (B z ) for TM (TE) waves. The scalar function y> satisfies the
twodimensional wave equation (8.19):
(V, 2 + y 2 )y = (8.34)
where
CO 2
y 2 = /* —  k 2 (8.35)
subject to the boundary condition :
dip
for TM (TE) waves.
y> \ s = 0, or
on
= (8.36)
[Sect. 8.3]
Wave Guides and Resonant Cavities
245
Equation (8.34) for y>, together with boundary condition (8.36), specifies
an eigenvalue problem. It is easy to see that the constant y 2 must be non
negative. Roughly speaking, it is because y must be oscillatory in order
to satisfy boundary condition (8.36) on opposite sides of the cylinder.
There will be a spectrum of eigenvalues y x 2 and corresponding solutions
y x , A = 1, 2, 3, . . . , which form an orthogonal set. These different
solutions are called the modes of the guide. For a given frequency a>, the
wave number k is determined for each value of A:
,2 w 2
(8.37)
If we define a cutoff frequency a> x ,
7i
then the wave number can be written:
k 2 =
\/ fX€\J CO 2 — Oi x
(8.38)<
(8.39)<
We note that, for a> > co x , the wave number k x is real; waves of the A
mode can propagate in the guide. For frequencies less than the cutoff
frequency, k x is imaginary; such modes cannot propagate and are called
cutoff modes. The behavior of the axial wave number as a function of
frequency is shown qualitatively in Fig. 8.4. We see that at any given
frequency only a finite number of modes can propagate. It is often con
venient to choose the dimensions of the guide so that at the operating
frequency only the lowest mode can occur. This is shown by the vertical
arrow on the figure.
Since the wave number k x is always less than the freespace value
VJuco/c, the wavelength in the guide is always greater than the freespace
Fig. 8.4 Wave number k x versus
frequency o> for various modes A.
co x is the cutoff frequency.
0>4 «5
246
Classical Electrodynamics
wavelength. In turn, the phase velocity v v is larger than the infinite space
value :
_ o> _ _c 1 c
The phase velocity becomes infinite exactly at cutoff.
(8.40)
8.4 Modes in a Rectangular Wave Guide
As an important illustration of the general features described in
Section 8.3 we consider the propagation of TE waves in a rectangular wave
guide with inner dimensions a, h, as shown in Fig. 8.5. The wave equation
for y) = B z is
/ a 2 a 2 ,\
(8.41)
with boundary conditions dxpjdn = at x = 0, a and y = 0, b. The
solution for ip is consequently
Vmn(*» V) = B cos iTm J cos (UI^j ( 8 42)
where
\ a 2 bV
(8.43)
The single index X specifying the modes previously is now replaced by the
two positive integers m, n. In order that there be nontrivial solutions, m
and n cannot both be zero. The cutoff frequency co mn is given by
co„
= [c~\^—(— 4 — V
Jue\a 2 b 2 /
(8.44)"
Fig. 8.5
[Sect. 8.4]
Wave Guides and Resonant Cavities
247
If a > b, the lowest cutoff frequency, that of the dominant TE mode,
occurs for m = 1, n = 0:
c 10 = p (8.45)
This corresponds to onehalf of a freespace wavelength across the guide.
The explicit fields for this mode, denoted by TE 10 , are:
B„
= B n cos\—)e ikz 
ika
B = B n sin
1 7TX \ ikzicot
(8.46)
ttc \ a I
The presence of a factor i in B x (and E y ) means that there is a spatial (or
temporal) phase difference of 90° between B x (and E y ) and B, in the
propagation direction. It happens that the TE X mode has the lowest
cutoff frequency of both TE and TM modes,* and so is the one used in
most practical situations. For a typical choice a = 2b the ratio of cutoff
frequencies co mn for the next few modes to a> 10 are as follows:
n
►
1
2
3
2.00
4.00
6.00
1
1.00
2.24
4.13
m 2
2.00
2.84
4.48
I 3
3.00
3.61
5.00
4
4.00
4.48
5.66
5
5.00
5.39
6
6.00
There is a frequency range from cutoff to twice cutoff where the TB X
mode is the only propagating mode. Beyond that frequency other modes
rapidly begin to enter. The field configurations of the TEj mode and
other modes are shown in many books, e.g., American Institute of Physics
Handbook, McGrawHill, New York (1957), p. 561.
* This is evident if we note that for the TM modes E z is of the form
~) sin \t)
while y 2 is still given by (8.43). The lowest mode has m = n = 1. Its cutoff frequency is
greater than that of the TE 1>0 mode by the factor 11+ — 1 .
248
Classical Electrodynamics
8.5 Energy Flow and Attenuation in Wave Guides
The general discussion of Section 8.3 for a cylindrical wave guide of
arbitrary crosssectional shape can be extended to include the flow of
energy along the guide and the attenuation of the waves due to losses in the
walls having finite conductivity. The treatment will be restricted to one
mode at a time; degenerate modes will be mentioned only briefly. The
flow of energy is described by the complex Poynting's vector:
S =
c
Air.
(Ex H*)
(8.47)*
whose real part gives the timeaveraged flux of energy. For the two types
of field we find, using (8.24):
S =
cok
e
L k J
tory*
1
v 2
e 3 IV^I 2  i y — y*V t xp
k J
(8.48)
where the upper (lower) line is for TM (TE) modes. Since y> is generally
real,* we see that the transverse component of S represents reactive energy
flow and does not contribute to the timeaverage flux of energy. On the
other hand, the axial component of S gives the timeaveraged flow of
energy along the guide. To evaluate the total power flow P we integrate
the axial component of S over the crosssectional area A :
Js.
Ja
e, da =
■W)* m Wti)da
(8.49)
By means of Green's first identity (1.34) applied to two dimensions, (8.49)
can be written:
P =
i>y>*^dl \y>*V 2 yda
LJc dn Ja
(8.50)
where the first integral is around the curve C which defines the boundary
surface of the cylinder. This integral vanishes for both types of fields
* It is possible to excite a guide in such a manner that a given mode or linear combina
tion of modes has a complex xp. Then a timeaveraged transverse energy flow can occur.
Since it is a circulatory flow, however, it really only represents stored energy and is not
of great practical importance.
[Sect. 8.5] Wave Guides and Resonant Cavities 249
because of boundary conditions (8.36). By means of the wave equation
(8.34) the second integral may be reduced to the normalization integral for
ip. Consequently the transmitted power is
P =
£M^ m ML^° ^
where the upper (lower) line is for TM (TE) modes, and we have exhibited
all the frequency dependence explicitly.
It is straightforward to calculate the field energy per unit length of the
guide in the same way as the power flow. The result is
u =[t] l 2^ML^ da (8  52) *
Comparison with the power flow P shows that P and U are proportional.
The constant of proportionality has the dimensions of velocity (velocity
of energy flow) and is just the group velocity :
£_*£!' ^/lSi.,. (8.53)
U co jue ^Jfic
as can be verified by a direct calculation of v g = dco/dk from (8.39),
assuming that the dielectric filling the guide is nondispersive. We note
that v g is always less than the velocity of waves in an infinite medium and
falls to zero at cutoff. The product of phase velocity (8.40) and group
velocity is constant:
v p v g =  (8.54)
an immediate consequence of the fact that co Aco oc k AA:.
Our considerations so far have applied to wave guides with perfectly
conducting walls. The axial wave number k x was either real or purely
imaginary. If the walls have a finite conductivity, there will be ohmic
losses and the power flow along the guide will be attenuated. For walls
with large conductivity the wave number will have a small imaginary part:
fc.fcf + i^ (8.55)*
where k x 0) is the value for perfectly conducting walls. The attenuation
constant /3 A can be found either by solving the boundaryvalue problem
over again with boundary conditions appropriate for finite conductivity,
or by calculating the ohmic losses by the methods of Section 8.1 and
250
Classical Electrodynamics
using conservation of energy. We will use the latter technique. The
power flow along the guide will be given by
P(z) = P e~ 2 ^ z
Thus the attenuation constant is given by
1 dP
&=
IP dz
(8.56)"
(8.57)*
where —dP/dz is the power dissipated in ohmic losses per unit length of the
guide. According to the results of Section 8.1, this power loss is
dP
dz
1
I67T J2adjbf Jc
In x Bl 2 dl
(8.58)*
where the integral is around the boundary of the guide. With fields (8.32)
and (8.33) it is easy to show that for a given mode:
dP
dz ZlTpad/jL
'\coJ Jc
I jU€CO x
dtp
dn
^) n xV (
co 2 /
vl 2 + ^rM 2
Ydl
(8.59)
where again the upper (lower) line applies to TM (TE) modes.
Since the transverse derivatives of xp are determined entirely by the size
and shape of the wave guide, the frequency dependence of the power loss
is explicitly exhibited in (8.59). In fact, the integrals in (8.59) may be simply
estimated from the fact that for each mode :
V, 2 + ^Hy, =
(8.60)
This means that, in some average sense, and barring exceptional circum
stances, the transverse derivatives of ip must be of the order of magnitude
of Vjbt€(a)Jc)ip :
dip
dn
CO
(\nxVM 2 )^^vW)
(8.61)
Consequently the line integrals in (8.59) can be related to the normalization
integral of y; 2 over the area. For example,
cco x ~
dtp
dn
AJa
dl = l^e— I \ip\ 2 da
AJa
(8.62)
where C is the circumference and A is the area of cross section, while g x is
a dimensionless number of the order of unity. Without further knowledge
[Sect. 8.5]
Wave Guides and Resonant Cavities
251
of the shape of the guide we can obtain the order of magnitude of the
attenuation constant fi x and exhibit completely its frequency dependence.
Thus, using (8.59) with (8.62) and (8.51), plus the frequency dependence of
the skin depth (7.85), we find
fti
c
.477.
.±(<L\
ft od x \2A/
\ or 1
£i + Vx
Oil 
(8.63)"
where a is the conductivity (assumed independent of frequency), d x is the
skin depth at the cutoff frequency, and £ A , r\ x are dimensionless numbers
of the order of unity. For TM modes, r\ x = 0.
For a given crosssectional geometry it is a straightforward matter to
calculate the dimensionless parameters £ A and r\ x in (8.63). For the TE
modes with n = in a rectangular guide, the values are  w0 = a/ (a + b)
and r} mfi = 2b\{a + b). For reasonable relative dimensions, these
parameters are of order unity, as expected.
t
0A
_
\
TM^
—
— J£— —
1
IjH 1
1 1
»/«>
Fig. 8.6 Attenuation constant $\
as a function of frequency for
typical TE and TM modes. For
TM modes the minimum atten
uation occurs at co/cox = "^3, re
gardless of crosssectional shape.
The general behavior of /S A as a function of frequency is shown in
Fig. 8.6. Minimum attenuation occurs at a frequency well above cutoff.
For TE modes the relative magnitudes of t x and rj x depend on the shape
of the guide and on X. Consequently no general statement can be made
about the exact frequency for minimum attenuation. But for TM modes
the minimum always occurs at co min = V3oj x . At high frequencies the
attenuation increases as co 1 ^. In the microwave region typical attenuation
constants for copper guides are of the order /9 A ~ lOr^coJc, giving \fe
distances of 200400 meters.
The approximations employed in obtaining (8.63) break down close to
cutoff. Evidence for this is the physically impossible, infinite value of
(8.63) at a> = co x . A treatment of the problem by perturbation theory
252 Classical Electrodynamics
with the boundary condition (8.11) yields the more accurate result,
k * = k <m + 2(1 + i)k (o)^ (8>64)
where & is still given by (8.63) For k[ 0) > X this reduces to our previous
expression (8.55). But at cutoff (k{ 0) = 0) the wave number is now finite
with real and imaginary parts of the order of the geometrical mean of
ojJc and a typical value of fj x , say at co ~ 2co A .
In the discussion so far we have considered only one mode at a time.
This procedure fails whenever a TE and a TM mode have the same cutoff
frequency, as occurs in the rectangular guide, for example, with n ^ 0,
m ^ 0. The reason for the failure is that the boundary condition (8.11)
for finite conductivity couples the degenerate modes. The calculation of
the attenuation then involves socalled degenerate state perturbation
theory, and the expression for £ takes the form,
P = KAtm + Ate) ± A/(£ TM  £ TE ) 2 + tf* (8.65)
where /9 TM and /? TB are the values found above, while K is a coupling
parameter. The two values of ft in (8.65) give the attenuation for the
two orthogonal, mixed modes which satisfy the perturbed boundary
conditions. *
8.6 Resonant Cavities
Although an electromagnetic cavity resonator can be of any shape
whatsoever, an important class of cavities is produced by placing end
faces on a length of cylindrical wave guide. We will assume that the end
surfaces are plane and perpendicular to the axis of the cylinder. As usual,
the walls of the cavity are taken to have infinite conductivity, while the
cavity is filled with a lossless dielectric with constants ju, e. Because of
reflections at the end surfaces the z dependence of the fields will be that
appropriate to standing waves :
A sin kz + B cos kz
If the plane boundary surfaces are at z = and z = d, the boundary
conditions can be satisfied at each surface only if
k = Pl> P = 0,1,2,... (8.66)
* For the theory of perturbation of boundary conditions in guides and cavities, see
G. Goubau, Electromagnetic Waveguides and Cavities, Pergamon Press, New York,
1961; Sect. 25. Attenuation for degenerate modes in guides is treated by R. Muller,
Z. Naturforsch., 4a, 218 (1949), and for the rectangular cavity by the same author in
Sect. 37 of the book by Goubau.
[Sect. 8.6] Wave Guides and Resonant Cavities 253
For TM fields the vanishing of E t at z = and z = d requires
E z = y>{x, y) cos (^), p = 0, 1, 2, . . . (8.67)
Similarly for TE fields, the vanishing of B z at z = and z = d requires
B z = y>(x, y) sin (^), p = 1, 2, 3, . . . (8.68)
Then from (8.24) we find the transverse fields :
TM FIELDS
(8.69)
TE FIELDS
E^^sin^^xV^
B( = £Lcos(^W
' dy 2 \ d 1
(8.70)
The boundary conditions at the ends of the cavity are now explicitly
satisfied. There remains the eigenvalue problem (8.34)(8.36), as before.
But now the constant y 2 is :
/
= "<7(t) 2 (8  71)
For each value of p the eigenvalue y x 2 determines an eigenfrequency of
resonance frequency a) Xp :
d
l v = ^ J \y, 2 +( E jJ] (»72)«
and the corresponding fields of that resonant mode. The resonance
frequencies form a discrete set which can be determined graphically on the
figure of axial wave number k versus frequency in a wave guide (see p. 245)
by demanding that k = prr/d. It is usually expedient to choose the
various dimensions of the cavity so that the resonant frequency of operation
lies well separated from other resonant frequencies. Then the cavity will
be relatively stable in operation and insensitive to perturbing effects
associated with frequency drifts, changes in loading, etc.
254
Classical Electrodynamics
Fig. 8.7
An important practical resonant cavity is the right circular cylinder,
perhaps with a piston to allow tuning by varying the height. The cylinder
is shown in Fig. 8.7, with inner radius R and length d. For a TM mode
the transverse wave equation for y> = E z , subject to the boundary con
dition E g = at p = R, has the solution :
where
V(P, <f>) = J m (ymnP)e ±im *
Ymn
R
(8.73)
(8.74)
x mn is the «th root of the equation, J m (x) = 0. These roots are given on
page 72, below equation (3.92). The integers m and n take on the values
m — 0, 1, 2, . . . , and n = 1, 2, 3, . . . . The resonance frequencies are
given by
^' tmin/n ^^
V/" €
X mn , P iP
R 2 d 2
(8.75)*
The lowest TM mode has m = 0, n = 1, p = 0, and so is designated
TM 010 . Its resonance frequency is
_ 2.405 c
^010
COnin —
The explicit expressions for the fields are
(8.76)
(8.77)
The resonant frequency for this mode is independent of d. Consequently
simple tuning is impossible.
[Sect. 8.7] Wave Guides and Resonant Cavities 255
For TE modes, the basic solution (8.73) still applies, but the boundary
condition on B,
«.•>
makes
(8.78)
R
where x' mn is the «th root of J m '(x) = 0. These roots, for a few values of
m and n, are tabulated below :
Roots of J m '(x) =
m =
m = 1
m = 2
m = 3
x' 0n = 3.832, 7.016, 10.174,
x' ln = 1.841, 5.331, 8.536,
x' 2n = 3.054, 6.706, 9.970,
x' Zn = 4.201, 8.015, 11.336,
The resonance frequencies are given by
„„..Ii/^. + £^ (8 . 7 9).
tymv
where m = 0, 1, 2, ... , but n,p = 1, 2, 3, ... . The lowest TE mode has
m = n = p = I, and is denoted TE X x x . Its resonance frequency is
_ 1.841 c(
V^M 1 + 2 ' 912 ^ (880)
while the fields are derivable from
B 2 = Vi^i^f) cos ^ sin {^)e i(0t (8.81)
by means of (8.70). For d large enough (d > 2.03/J), the resonance
frequency eo m is smaller than that for the lowest TM mode (8.76). Then
the TE ljlfl mode is the fundamental oscillation of the cavity. Because the
frequency depends on the ratio d/R it is possible to provide easy tuning by
making the separation of the end faces adjustable.
8.7 Power Losses in a Cavity; Q of a Cavity
In the preceding section it was found that resonant cavities had discrete
frequencies of oscillation with a definite field configuration for each
resonance frequency. This implies that, if one were attempting to excite a
particular mode of oscillation in a cavity by some means, no fields of the
256 Classical Electrodynamics
right sort could be built up unless the exciting frequency were exactly equal
to the chosen resonance frequency. In actual fact there will not be a delta
function singularity, but rather a narrow band of frequencies around the
eigenfrequency over which appreciable excitation can occur. An important
source of this smearing out of the sharp frequency of oscillation is the
dissipation of energy in the cavity walls and perhaps in the dielectric filling
the cavity. A measure of the sharpness of response of the cavity to external
excitation is the Q of the cavity, defined as 2tt times the ratio of the
timeaveraged energy stored in the cavity to the energy loss per cycle:
Q _ ^ Stored energy
Power loss
Here co is the resonance frequency, assuming no losses. By conservation
of energy the power dissipated in ohmic losses is the negative of the time
rate of change of stored energy U. Thus from (8.82) we can write an
equation for the behavior of U as a function of time:
dU _ cop T j
dt ~ Q
with solution
U(t) = Uoe^^
(8.83)
If an initial amount of energy U is stored in the cavity, it decays away
exponentially with a decay constant inversely proportional to Q. The
time dependence in (8.83) implies that the oscillations of the fields in the
cavity are damped as follows :
£(0 = Eoe o> t/2Q e i«>ot (8.84)
A damped oscillation such as this has not a pure frequency, but a super
position of frequencies around co — co . Thus,
1 f 00
£(0 = ±= E(co)e 1(0t dco
JllT J oo
4
where
1 f 00
E(C0) = = E <oot/2Q e Ha><oo)t df
yjllT JO
(8.85)
The integral in (8.85) is elementary and leads to a frequency distribution
for the energy in the cavity having a Lorentz line shape :
£(w)2 K ( v!w /?m« (8  86)
(co  w Q y + (co /2Q)
[Sect. 8.7]
Wave Guides and Resonant Cavities
257
The resonance shape (8.86), shown in Fig. 8.8, has a full width at half
maximum (confusingly called the half width) equal to coJQ. For a
constant input voltage, the energy of oscillation in the cavity as a function
of frequency will follow the resonance curve in the neighborhood of a
particular resonant frequency. Thus, if Aco is the frequency separation
between halfpower points, the Q of the cavity is
2 = ?
Aco
(8.87)
Q values of several hundreds or thousands are common for microwave
cavities.
To determine the Q of a cavity we must calculate the timeaveraged
energy stored in it and then determine the power loss in the walls. The
computations are very similiar to those done in Section 8.5 for attenuation
in wave guides. We will consider here only the cylindrical cavities of
Section 8.6, assuming no degeneracies (see the footnote on p. 252). The
energy stored in the cavity for the mode A, p is, according to (8.67)(8.70) :
where the upper (lower) line applies to TM (TE) modes. For the TM
modes with p = the result must be multiplied by 2.
The power loss can be calculated by a modification of (8.58):
p >~=[£\Mi d i
dz\n x Blfides + 2 da\n x B nd8
> j da\n x B nd8 ]
(8.89)"
For TM modes with p ^ it is easy to show that
Fig. 8.8 Resonance line shape. The
full width Aco at halfmaximum (of
the power) is equal to the central
frequency eo divided by the Q of the
cavity.
coJQ = Aco
258 Classical Electrodynamics
where the dimensionless number $ x is the same one that appears in (8.62),
C is the circumference of the cavity, and A is its crosssectional area. For
p = 0, £ A must be replaced by 2l x . Combining (8.88) and (8.89) according
to (8.82), and using definition (7.85) for the skin depth d, we find the Q of
the cavity :
_£d 1
Q= */ 77: (8  91) "
AA>
where fx c is the permeability of the metal walls of the cavity. For p =
modes, (8.91) must be multiplied by 2 and £ x replaced by 21 x . This
expression for Q has an intuitive physical interpretation when written in
the form :
Q = ^(— ) x (Geometrical factor) (8.92)*
where Fis the volume of the cavity, and S its total surface area. The Q of
a cavity is evidently, apart from a geometrical factor, the ratio of the
volume occupied by the fields to the volume of the conductor into which
the fields penetrate because of the finite conductivity. For TM modes in
cylindrical cavities the geometrical factor is
(8.93)
for p =£ 0, and is
211 +mL
' + *§)
AA)
Cd\
(8.94)
for/? = modes. For TE modes in the cylindrical cavity the geometrical
factor is somewhat more complicated, but of the same order of magnitude.
For the TM 01>0 mode in a circular cylindrical cavity with fields (8.77),
 A = 1 (true for all TM modes), so that the geometrical factor is 2 and
Q is:
(■♦£)
[Sect. 8.8] Wave Guides and Resonant Cavities 259
For the TE lfl>1 mode calculation yields a geometrical factor*
K)
(l + 0.209  + 0.242 4 1
V R R 3 /
(8.96)
and a Q :
( i+o  344 S)
^ ^ (1 + 0.209  + 0.242 — )
\ R RV
e^iTb — : ; — "' „x ( 8  97 )'
Expression (8.92) for g applies not only to cylindrical cavities but also
to cavities of arbitrary shape, with an appropriate geometrical factor of
the order of unity.
8.8 Dielectric Wave Guides
In Sections 8.28.5 we considered wave guides made of hollow metal
cylinders with fields only inside the hollow. Other guiding structures are
possible. The parallelwire transmission line is an example. The general
requirement for a guide of electromagnetic waves is that there be a flow of
energy only along the guiding structure and not perpendicular to it. This
means that the fields will be appreciable only in the immediate neighbor
hood of the guiding structure. For hollow wave guides these requirements
are satisfied in a trivial way. But for an open structure like the parallel
wire line the fields extend somewhat away from the conductors, falling off
like p~ 2 for the TEM mode, and exponentially for higher modes.
A dielectric cylinder, such as shown in Fig. 8.9, can serve as a wave guide,
with some properties very similar to those of a hollow metal guide if its
dielectric constant is large enough. There are, however, characteristic
differences which arise because of the very different boundary conditions
to be satisfied at the surface of the cylinder. The general considerations of
Section 8.2 still apply, except that the transverse behavior of the fields is
governed by two equations like (8.19), one for inside the cylinder and one
for outside :
INSIDE
V *'+(^ e i£ **)]{§ = ° < 8  98 >
* Note that this factor varies by only 30 per cent as the cylinder geometry is changed
from djR^>\ to d/R < 1.
260
Classical Electrodynamics
OUTSIDE
V+\wo^k*
=
(8.99)
Both dielectric (/f l5 ej) and surrounding medium (/u , e ) are assumed to be
uniform and isotropic in their properties. The axial propagation constant
k must be the same inside and outside the cylinder in order to satisfy
boundary conditions at all points on the surface at all times.
In the usual way, inside the dielectric cylinder the transverse Laplacian
of the fields must be negative so that the constant
V = /"i € i
k 2
(8.100)
is positive. Outside the cylinder, however, the requirement of no transverse
flow of energy demands that the fields fall off exponentially. (There is no
TEM mode for a dielectric guide.) Consequently, the quantity in (8.99)
equivalent to y 2 must be negative. Therefore we define a quantity /? 2 :
£ 2 = k 2  fi e —
c l
(8.101)
and demand that acceptable wave guide solutions have /3 2 positive (/? real).
The oscillatory solutions (inside) must be matched to the exponential
solutions (outside) at the boundary of the dielectric cylinder. The
boundary conditions are continuity of normal B and D and tangential E
and H, rather than the vanishing of normal B and tangential E (8.25)
appropriate for hollow conductors. Because of the more involved
boundary conditions the types of fields do not separate into TE and TM
modes, except in special circumstances such as azimuthal symmetry in
Moeo
Fig. 8.9 Section of dielectric wave
guide.
[Sect. 8.8]
Wave Guides and Resonant Cavities
261
circular cylinders, to be discussed below. In general, axial components
of both E and B exist. Such waves are sometimes designated as HE
modes.
To illustrate some of the features of the dielectric wave guide we consider
a circular cylinder of radius a consisting of nonpermeable dielectric with
dielectric constant e x in an external nonpermeable medium with dielectric
constant e . As a simplifying assumption we take the fields to have no
azimuthal variation. Then in cylindrical coordinates the radial equations
for E z or B z are Bessel's equations :
d 2 , 1 d , o . _ ^
dp p dp
jL + iA 0*^ = 0, P >a
dp p dp
(8.102)
The solution, satisfying the requirements of finiteness at the origin and at
infinity, is found from Section 3.6 to be:
V = "j
I Jo(yp)>
AKtfp),
(8.103)
The other components of E and B can be found from (8.24) when the
relative amounts of E z and B z are known. With no <f> dependence to the
fields, (8.24) reduces to
INSIDE
ik dB,
D ie^ dE z
y l c dp
E * = ~fk B "
€iCO
(8.104)
and similar expressions for p > a. The fact that the fields arrange them
selves in two groups, (B p , Ej) depending on B z , and (B^, E p ) depending on
E z , suggests that we attempt to obtain solutions of the TE or TM type, as
for the metal wave guides. For the TE modes, the fields are explicitly
B z = UYP)
y
cy
p < a
(8.105)
262
and
Classical Electrodynamics
B z = AK (P P )
p
p > a
(8.106)
These fields must satisfy the standard boundary conditions at p = a. This
leads to the two conditions,
AK (fia) = J (ya)
p y
(8.107)
Upon elimination of the constant A we obtain the determining equations
for y, (3, and therefore k:
Uya)  K^a) = Q
(8.108)
yJ (ya) ^K (^a)
and, from (8.100) and (8.101),
r 2 + ^ = ( ei e )^
c l
The general behavior of the two parts of the first equation in (8.108) is
shown in Fig. 8.10a. Figure 8.106 shows the two curves superposed
Pa
(Ta)max =
(a)
(b)
Fig. 8.10 Graphical determination of the axial propagation constant for a
dielectric wave guide.
[Sect. 8.8] Wave Guides and Resonant Cavities 263
according to the second equation in (8.108). The frequency is assumed, to
be high enough that two modes, marked by the circles at the intersections
of the two curves, exist. The vertical asymptotes are given by the roots of
J Q (x) = 0. If the maximum value of ya is smaller than the first root
Ooi = 2.405), there can be no intersection of the two curves for real £.
Hence the lowest "cutoff" frequency for TE n waves is given by
2.405c
w i = , (8.109)
V e i _ e o a
At this frequency £ 2 = 0, but the axial wave number k is still real and
equal to its freespace value A/e w/c. Immediately below this "cutoff"
frequency, the system no longer acts as a guide but as an antenna, with
energy being radiated radially. For frequencies well above cutoff, ft and
k are of the same order of magnitude and are large compared to y provided
€ ± and e are not nearly equal.
For TM modes, the first equation in (8.108) is replaced by
yJ (ya) ^fiK^fia) V ' }
It is evident that all the qualitative features shown in Fig. 8.10 are retained
for the TM waves. The lowest "cutoff" frequency for TM „ waves is
clearly the same as for TE >n waves. For e x > e , provided the maximum
value of ya does not fall very near one of the roots of J (x) = 0, (8.110)
shows that the propagation constants are determined by J x {ya) ~ 0. This
is just the determining equation for TE waves in a metallic wave guide.
The reason for the equivalence of the TM modes in a dielectric guide and
the TE modes in a hollow metallic guide can be traced to the symmetry
of Maxwell's equations under the interchange of E and B (with appro
priate sign changes and factors of Vjue), plus the correspondence between
the vanishing of normal B at the metallic surface and the almost vanishing
of normal E at the dielectric surface (due to continuity of normal D with
If e x > e , then from (8.100) and (8.101) it is clear that the outside decay
constant is much larger than y, except near cutoff. This means that the
fields do not extend appreciably outside the dielectric cylinder. Figure
8. 1 1 shows qualitatively the behavior of the fields for the TE 0jl mode. The
other modes behave similarly. As mentioned earlier, modes with azimuthal
dependence to the fields have longitudinal components of both E and B.
Although the mathematics is somewhat more involved (see Problem 8.6),
the qualitative features of propagation— short wavelength along the
cylinder, rapid decrease of fields outside the cylinder, etc.— are the same
as for the circularly symmetric modes.
264
Classical Electrodynamics
a
1
1
— ^ 1
B,
E
B
/b p
X I
\ 1
\ 1
\ 1
\ 1
\l
E <t> \
1
1
p
— >
Fig. 8.11 Radial variation of
fields of TE ,i mode in dielectric
guide. For €i ^> e , the fields
are confined mostly inside the
dielectric.
Dielectric wave guides have not been used for microwave propagation,
except for special applications. One reason is that it is difficult to obtain
suitable dielectrics with sufficiently low losses at microwave frequencies.
In a recent application at optical frequencies very fine dielectric filaments,
each coated with a thin layer of material of much lower index of refraction,
are closely bundled together to form imagetransfer devices.* The
filaments are sufficiently small in diameter (~ 10 microns) that waveguide
concepts are useful, even though the propagation is usually a mixture of
several modes.
REFERENCES AND SUGGESTED READING
Wave guides and resonant cavities are dealt with in numerous electrical and communi
cations engineering books. Among the physics textbooks which discuss guides, trans
mission lines, and cavities are
Panofsky and Phillips, Chapter 12,
Slater,
Sommerfeld, Electrodynamics, Sections 2225,
Stratton, Sections 9.189.22.
The mathematical tools for the discussion of these boundaryvalue problems are
presented by
Morse and Feshbach, especially Chapter 13.
Information on special functions may be found in the everreliable
Magnus and Oberhettinger.
Numerical values of Bessel functions are given by
Jahnke and Emde,
Watson.
PROBLEMS
8.1 A transmission line consisting of two concentric circular cylinders of metal
with conductivity a and skin depth <5, as shown on p. 265, is filled with a
* B. O'Brien, Physics Today, 13, 52 (1960).
[Probs. 8]
Wave Guides and Resonant Cavities
265
uniform lossless dielectric (ji, e). A TEM mode is propagated along this line.
(a) Show that the timeaveraged power flow along the line is
P =
ira 2 \H n \ 2 In
9
where H is the peak value of the azimuthal magnetic field at the surface of
the inner conductor.
(6) Show that the transmitted power is attenuated along the line as
P(z) =p e2rz
where
2ct<5
M , (b\
In I I
(c) The characteristic impedance Z of the line is defined as the ratio of
the voltage between the cylinders to the axial current flowing in one of them
at any position z. Show that for this line
_ c _\2tt/\] e \aj
(d) Show that the series resistance and inductance per unit length of the
line are
2tto6 \a bj
2t7 \a/ 4tt \a b
where n c is the permeability of the conductor. The correction to the
inductance comes from the penetration of the flux into the conductors by a
distance of order 8.
8.2 A transmission line consists of two identical thin strips of metal, shown in
cross section on p. 266. Assuming that b > a, discuss the propagation
266 Classical Electrodynamics
of a TEM mode on this line, repeating the derivations of Problem 8.1.
Show that _ _ _
ab U. 2
p
c
Att_
c
v
4tt
4tt
o
c
aod/d ju
Itia
Ab
R =
2adb
L =
/ fig + /j, c d \
\ b J
M J
where the symbols have the same meanings as in Problem 8.1.
8.3 Transverse electric and magnetic waves are propagated along a hollow,
right circular cylinder of brass with inner radius R.
(a) Find the cutoff frequencies of the various TE and TM modes. Deter
mine numerically the lowest cutoff frequency (the dominant mode) in terms
of the tube radius and the ratio of cutoff frequencies of the next four higher
modes to that of the dominant mode.
(b) Calculate the attenuation constant of the wave guide as a function of
frequency for the lowest two modes and plot it as a function of frequency.
8.4 A wave guide is constructed so that the cross_ section of the guide forms a
right triangle with sides of length a, a, V2a, as shown on p. 267. The
medium inside has /u = e = 1 .
[Probs. 8]
Wave Guides and Resonant Cavities
267
(a) Assuming infinite conductivity for the walls, determine the possible
modes of propagation and their cutoff frequencies.
(b) For the lowest modes of each type calculate the attenuation constant,
assuming that the walls have large, but finite, conductivity. Compare the
result with that for a square guide of side a made from the same material.
8.5 A resonant cavity of copper consists of a hollow, right circular cylinder of
inner radius R and length L, with flat end faces.
(a) Determine the resonant frequencies of the cavity for all types of
waves. With (c/V/x € R) as a unit of frequency, plot the lowest four resonant
frequencies of each type as a function of RjL for < RjL < 2. Does the
same mode have the lowest frequency for all RjLl
(b) If R = 2 cm, L = 3 cm, and the cavity is made of pure copper, what
is the numerical value of Q for the lowest resonant mode ?
8.6 A right circular cylinder of nonpermeable dielectric with dielectric constant e
and radius a serves as a dielectric wave guide in vacuum.
(a) Discuss the propagation of waves along such a guide, assuming that
the azimuthal variation of the fields is e im ^.
(b) For m = ±1, determine the mode with the lowest cutoff frequency
and discuss the properties of its fields (cutoff frequency, spatial variation,
etc.), assuming that e > 1 .
9
Simple Radiating Systems
and Diffraction
In Chapters 7 and 8 we have discussed the properties of electro
magnetic waves and their propagation in both bounded and unbounded
geometries. But nothing has been said about how to produce these waves.
In the present chapter we remedy this omission to some extent by pre
senting a discussion of radiation by a localized oscillating system of
charges and currents. The treatment is straightforward, with little in the
way of elegant formalism. It is by its nature restricted to rather simple
radiating systems. A more systematic approach to radiation by localized
distributions of charge and current is left to Chapter 16, where multipole
fields are discussed.
The second half of the chapter is devoted to the subject of diffraction.
Since the customary scalar Kirchhoff theory is discussed in many books,
the emphasis has been placed on the vector properties of the electro
magnetic field in diffraction.
9.1 Fields and Radiation of a Localized Oscillating Source
For a system of charges and currents varying in time we can make a
Fourier analysis of the time dependence and handle each Fourier com
ponent separately. We therefore lose no generality by considering the
potentials, fields, and radiation from a localized system of charges and
currents which vary sinusoidally in time :
P (x, = P (x)e^ (9 1}
j(x, = J(x)e" J
268
[Sect. 9.1] Simple Radiating Systems and Diffraction 269
As usual, the real part of such expressions is to be taken to obtain physical
quantities. The electromagnetic potentials and fields are assumed to have
the same time dependence.
It was shown in Chapter 6 that the solution for the vector potential
A(x, t) in the Lorentz gauge is
A(x, =  fdV [df J(X '' ° d(t' + x ~ x ' [  t] (9.2)
cJ J x — x' \ C I
provided no boundary surfaces are present. The Dirac delta function
assures the causal behavior of the fields. With the sinusoidal time
dependence (9.1), the solution for A becomes
I r e ik\xx'\
A(x) =  J(x') <Pz' (9.3)
cj x — x I
where k = ai/c is the wave number, and a sinusoidal time dependence is
understood. The magnetic induction is given by
B = V x A (9.4)
while, outside the source, the electric field is
E = V x B (9.5)
k
Given a current distribution J(x'), the fields can, in principle at least, be
determined by calculating the integral in (9.3). We will consider one or
two examples of direct integration of the source integral in Section 9.4.
But at present we wish to establish certain simple, but general, properties
of the fields in the limit that the source of current is confined to a small
region, very small in fact compared to a wavelength. If the source
dimensions are of order d and the wavelength is X = Ittc/oj, and if d < X,
then there are three spatial regions of interest:
The near (static) zone : d < r < X
The intermediate (induction) zone : d < r ^ X
The far (radiation) zone : d < X < r
We will see that the fields have very different properties in the different
zones. In the near zone the fields have the character of static fields with
radial components and variation with distance which depends in detail on
the properties of the source. In the far zone, on the other hand, the fields
are transverse to the radius vector and fall off as r _1 , typical of radiation
fields.
270 Classical Electrodynamics
For the near zone where r « X (or kr « 1) the exponential in (9.3) can
be replaced by unity. Then the vector potential is of the form already
considered in Chapter 5. The inverse distance can be expanded using
(3.70), with the result,
lim A(x) = i 2 T^TZ ^T^ \mr' l Yti?, <f>') d 3 *' (9.6)
fcro c t, to 2/ + 1 r + J
This shows that the near fields are quasistationary, oscillating har
monically as e~ i(ot , but otherwise static in character.
In the far zone {kr » 1) the exponential in (9.3) oscillates rapidly and
determines the behavior of the vector potential. In this region it is
sufficient to approximate
x — x' ~ r — n • x' (9.7)
where n is a unit vector in the direction of x. Furthermore, if only the
leading term in kr is desired, the inverse distance in (9.3) can be replaced
by r. Then the vector potential is
lim A(x) = — J(x>~ ifcn  X d z x'. (9.8)
at>oo cr J
This demonstrates that in the far zone the vector potential behaves as an
outgoing spherical wave. It is easy to show that the fields calculated
from (9.4) and (9.5) are transverse to the radius vector and fall off as r~ x .
They thus correspond to radiation fields. If the source dimensions are
small compared to a wavelength it is appropriate to expand the integral in
(9.8) in powers of A:: ikr / ;ky» f
lim A(x) = — 2 — — J(x')(n • x') B d*x' (9.9)
kr+oo Cr n n\ J
The magnitude of the nth term is given by
J(x')(fcn • x') n d*x' (9.10)
if.
Since the order of magnitude of x' is d and kd is small compared to unity
by assumption, the successive terms in the expansion of A evidently fall
off rapidly with n. Consequently the radiation emitted from the source
will come mainly from the first nonvanishing term in the expansion (9.9).
We will examine the first few of these in the following sections.
In the intermediate or induction zone the two alternative approxi
mations leading to (9.6) and (9.8) cannot be made; all powers of kr must
be retained. Without marshalling the full apparatus of vector multipole
fields, described in Chapter 16, we can abstract enough for our immediate
purpose. The key result is the exact expansion (16.22) for the Green's
function appearing in (9.3). For points outside the source (9.3) then
becomes »
A(*) = — 2 hf\kr)Y lm {B, +) J(x')j,(fcr')r,£(0', </>') d 3 x' (9.11)
C 7. TO J
Simple Radiating Systems and Diffraction 271
If the source dimensions are small compared to a wavelength, j t {kr') can
be approximated by (16.12). Then the result for the vector potential is of
the form of (9.6), but with the replacement,
^1 "* ^1 0 + a & kr> > + a ^ ikr ? +■'• + ^{ikrf ) (9.12)
The numerical coefficients a t come from the explicit expressions for the
spherical Hankel functions. The right hand side of (9.12) shows the
transition from the staticzone result (9.6) for kr « 1 to the radiationzone
form (9.9) for kr » 1.
9.2 Electric Dipole Fields and Radiation
If only the first term in (9.9) is kept, the vector potential is
ikr f
A(x) = 6 — J J(x') c/V (9.13)
Examination of (9.11) and (9.12) shows that (9.13) is the / = part of
the series and that it is valid everywhere outside the source, not just in the
far zone. The integral can be put in more familiar terms by an integration
by parts :
J J d z x' =  J x'(V . J) d z x' = ico\x'p(x') d 3 x' (9.14)
from the continuity equation,
iojp = V.j
Thus the vector potential is ., v ' }
r p tkr
where A(x)=f/cp (9 16)
P=jx' P (x')d 3 x' (9.17)
is the electric dipole moment, as defined in electrostatics by (4.8).
The electric dipole fields from (9.4) and (9.5) are
B = k 2 (n x p)— (l  —
r \ ikr
E = k\n x p) x n e ~ + [3n(n • p)  p](i  ^),
(9.18)
We note that the magnetic induction is transverse to the radius vector at
all distances, but that the electric field has components parallel and perpen
dicular to n.
In the radiation zone the fields take on the limiting forms,
a ikr\
B = fc 2 (n x p) —
E = B x n J
showing the typical behavior of radiation fields.
272 Classical Electrodynamics
In the near zone, on the other hand, the fields approach
B = ik(n x p) 
r 1
E=[3n(n.p)p]i
(9.20)
The electric field, apart from its oscillations in time, is just the static
electric dipole field (4.13). The magnetic induction is a factor (kr) smaller
than the electric field in the region where kr < 1 . Thus the fields in the
near zone are dominantly electric in nature. The magnetic induction
vanishes, of course, in the static limit k > 0. Then the near zone extends
to infinity.
The timeaveraged power radiated per unit solid angle by the oscillating
dipole moment p is
dP c
^ = ± Re [r 2 n . E x B*] (9.21)
dQ. 87T
where E and B are given by (9.19). Thus we find
dP c
^ =  fc*n x (n x p) 2 (9.22)
dil 67T
The state of polarization of the radiation is given by the vector inside the
absolute value signs. If the components of p all have the same phase, the
angular distribution is a typical dipole pattern,
^ = f fc 4 p 2 sin 2 (9.23)
dil 57T
where the angle 6 is measured from the direction of p. The total power
radiated is
P = C f IPI 2 (9.24)
A simple example of an electric dipole radiator is a centerfed, linear
antenna whose length d is small compared to a wavelength. The antenna
is assumed to be oriented along the z axis, extending from z = (d/2) to
z — —(d/2) with a narrow gap at the center for purposes of excitation, as
shown in Fig. 9.1. The current is in the same direction in each half of the
antenna, having a value I at the gap and falling approximately linearly to
zero at the ends :
I(z)e i0)t = lj\  'M\e i0>t (9.25)
[Sect. 9.3]
Simple Radiating Systems and Diffraction
273
Fig. 9.1 Short, centerfed, linear antenna.
From the continuity equation (9.15) the linearcharge density p (charge
per unit length) is constant along each arm of the antenna, with the value,
p'(*) = ±
2i7 fl
(9.26)
the upper (lower) sign being appropriate for positive (negative) values of z.
The dipole moment (9.17) is parallel to the z axis and has the magnitude
r
zp'(z) dz= 1 ^
(d/2) 2a>
The angular distribution of radiated power is
dP V
dQ. 327TC
while the total power radiated is
P
{kdf sin 2
I \kdf
12c
(9.27)
(9.28)
(9.29)
We see that for a fixed input current the power radiated increases as the
square of the frequency, at least in the longwavelength domain where
kd <1.
9.3 Magnetic Dipole and Electric Quadrupole Fields
The next term in expansion (9.9) leads to a vector potential,
A(x) = — (  ikj j* J(x')(n • x') dV (9.30)
where we have included the correct terms from (9.12) in order
to make (9.30) valid everywhere outside the source. This vector
potential can be written as the sum of two terms, one of which
gives a transverse magnetic induction and the other of which gives a
274 Classical Electrodynamics
transverse electric field. These physically distinct contributions can be
separated by writing the integrand in (9.30) as the sum of a part symmetric
in J and x' and a part that is antisymmetric. Thus
 (n • x')J = — [(n • x')J + (n • J)x'] + — (x' x J) x n (9.31)
c 2c 2c
The second, antisymmetric part is recognizable as the magnetization due
to the current J:
M = — (x x J) (9.32)
2c
The first, symmetric term will be shown to be related to the electric
quadrupole moment density.
Considering only the magnetization term, we have the vector potential,
A(x) = ik(n x m) — ( 1  — ) (9.33)
r \ ikrf
where m is the magnetic dipole moment,
m = \jl d z x = — (x x J) d 3 x (9.34)
The fields can be determined by noting that the vector potential (9.33) is
proportional to the magnetic induction (9.18) for an electric dipole. This
means that the magnetic induction for the present magnetic dipole source
will be equal to the electric field for the electric dipole, with the substitution
p —>■ m. Thus we find
B = k 2 (n x m) x n — + [3n(n • m)  m](   )e ikr (9.35)
r \r 3 r 2 /
Similarly, the electric field for a magnetic dipole source is the negative of
the magnetic field for an electric dipole :
E = k\n x m)— (l  — ) (9.36)
r \ ikrf
All the arguments concerning the behavior of the fields in the near and
far zones are the same as for the electric dipole source, with the inter
changes E>B, B— > — E, p^m. Similarly the radiation pattern and
total power radiated are the same for the two kinds of dipole. The only
difference in the radiation fields is in the polarization. For an electric
dipole the electric vector lies in the plane defined by n and p, while for a
magnetic dipole it is perpendicular to the plane defined by n and m.
[Sect. 9.3] Simple Radiating Systems and Diffraction 275
The integral of the symmetric term in (9.31) can be transformed by an
integration by parts and some rearrangement :
 f [(n • x')J + (n • J)x'] d*x' =  fx'(n • x>(x') d 3 x' (9.37)
The continuity equation (9.15) has been used to replace V • J by imp.
Since the integral involves second moments of the charge density, this
symmetric part corresponds to an electric quadrupole source. The vector
potential is
A(x) =   — (l  — ) f x'(n • x>(x') d 3 x' (9.38)
2 r \ ikrf J
The complete fields are somewhat complicated to write down. We will
content ourselves with the fields in the radiation zone. Then it is easy to
see that
B = ikn x A 1
(9.39)
E = ik(n x A) x n J
Consequently the magnetic induction is
* i 3 Hct /*
B =  — — (n x x')(n • x>(x') d 3 x' (9.40)
2 r J
With definition (4.9) for the quadrupole moment tensor,
Q* P = J(3*.*,  ^) P (x) d*x (9.41)
the integral in (9.40) can be written
n x J x'(n • x')p(x') d 3 x' = ^n x Q(n) (9.42)
The vector Q(n) is defined as having components,
<2a = I<Vv (943)
p
We note that it depends in magnitude and direction on the direction of
observation as well as on the properties of the source. With these defi
nitions we have the magnetic induction,
■r.3 Jkr
B =  — — n x Q(n) (9.44)
6 r
and the timeaveraged power radiated per unit solid angle,
— = — /c 6 [n x Q(n) 2 (9.45)
dQ. 288tt
276
Classical Electrodynamics
The general angular distribution is complicated. But the total power
radiated can be calculated in a straightforward way. With the definition of
Q(n) we can write the angular dependence as
n x Q(n) 2 = Q* • Q  n • Q 2
= J, Q*pQ« Y n p n y  2 Q*pQydn a n p n y n d
<*>/?, y,<5
(9.46)
The necessary angular integrals over products of the rectangular com
ponents of n are readily found to be
l
n R n y dQ
I
47T
477
J fiv
n a n p n y n d dQ. = — (d af} d yd + d ay d p5 + d ad d Py )
(9.47)
Then we find
/'
n x Q(n) 2 dQ = AnU 2 \Q a(S \ 2  A 2 Ql lQyy + ^2 \<L,
*,p
a,P
(9.48)
Since Q afi is a tensor whose main diagonal sum is zero, the first term in the
square brackets vanishes identically. Thus we obtain the final result for
the total power radiated by a quadrupole source:
'£2iw
(9.49)
The radiated power varies as the sixth power of the frequency for fixed
quadrupole moments, compared to the fourth power for dipole radiation.
A simple example of a radiating quadrupole source is an oscillating
spheroidal distribution of charge. The offdiagonal elements of Q aj3 vanish.
The diagonal elements may be written
(?33 — Qo> Qn — Q22 — — \Qo
Then the angular distribution of radiated power is
dP c/c 6
dQ. 128tt
Q ' sin 2 6 cos 2
(9.50)
(9.51)
This is a fourlobed pattern, as shown in Fig. 9.2, with maxima at 6 = tt/4
and 377/4. The total power radiated by this quadrupole is
P =
240
(9.52)
[Sect. 9.4]
Simple Radiating Systems and Diffraction
111
Fig. 9.2 Quadrupole radiation pattern.
The labor involved in manipulating higher terms in expansion (9.9) of
the vector potential (9.8) becomes increasingly prohibitive as the expansion
is extended beyond the electric quadrupole terms. Another disadvantage
of the present approach is that physically distinct fields such as those of the
magnetic dipole and the electric quadrupole must be disentangled from
the separate terms in (9.9). Finally, the present technique is useful only in
the long wavelength limit. A systematic development of multipole radia
tion is given in Chapter 16. It involves a fairly elaborate mathematical
apparatus, but the price paid is worth while. The treatment allows all
multipole orders to be handled in the same way; the results are valid for
all wavelengths ; the physically different electric and magnetic multipoles
are clearly separated from the beginning.
9.4 Centerfed Linear Antenna
For certain radiating systems the geometry of current flow is sufficiently
simple that integral (9.3) for the vector potential can be found in relatively
simple, closed form. As an example of such a system we consider a thin,
linear antenna of length d which is excited across a small gap at its mid
point. The antenna is assumed to be oriented along the z axis with its gap
at the origin, as indicated in Fig. 9.3. If damping due to the emission of
radiation is neglected, the current along the antenna can be taken as
sinusoidal in time and space with wave number k = co/c, and is symmetric
on the two arms of the antenna. The current vanishes at the ends of the
278 Classical Electrodynamics
antenna. Hence the current density can be written
J(x) = /sin(^/c Z )^)%)€ 3
(9.53)
for \z\ < (d/2). The delta functions assure that the current flows only
along the z axis. / is the peak value of the current if kd > tt. The current
at the gap is I = I sin (kd/2).
With the current density (9.53) the vector potential is in the z direction
and in the radiation zone has the form [from (9.7)] :
Ie 1 '
— ikz cos 6 j
e dz
kt , „ r f W2) . (kd .. A
A(x) = e 3 sin —  k \z\ I
cr J(,d/2) \ 2 1
The result of straightforward integration is
( cose) cos (I)
(9.54)
A(x) = €,
2Ie l
ckr 
cos
sin 2
(9.55)
Since the magnetic induction in the radiation zone is given by B =
ikn x A, its magnitude is B = k sin 6 \A 3 \. Thus the timeaveraged power
radiated per unit solid angle is
(kd A (kd\ 2
cos
dp
da
JL
2ttc
( cose) cos ()
sin 6
(9.56)
The electric vector is in the direction of the component of A perpendicular
to n. Consequently the polarization of the radiation lies in the plane
containing the antenna and the radius vector to the observation point.
Coaxial
feed
Fig. 9.3 Centerfed, linear antenna.
[Sect. 9.4]
Simple Radiating Systems and Diffraction
279
The angular distribution (9.56) depends on the value of kd. In the
long wavelength limit {kd < 1) it is easy to show that it reduces to the
dipole result (9.28). For the special values kd = tt (2tt), corresponding to
a half (two halves) of a wavelength of current oscillation along the antenna,
the angular distributions are
dP
dQ.
JL
lire
cos
(fees,)
sin
4 cos 4 I  cos
\2
sin 2
kd = tt
kd = 2TT
(9.57)
These angular distributions are shown in Chapter 16 in Fig. 16.4, where
they are compared to multipole expansions. The halfwave antenna
distribution is seen to be quite similar to a simple dipole pattern, but the
fullwave antenna has a considerably sharper distribution.
The fullwave antenna distribution can be thought of as due to the
coherent superposition of the fields of two halfwave antennas, one above
the other, excited in phase. The intensity at 6 = njl, where the waves add
algebraically, is 4 times that of a half wave antenna. At angles away from
6 = tt/2 the amplitudes tend to interfere, giving the narrower pattern. By
suitable arrangement of a set of basic antennas, such as the halfwave
antenna, with the phasing of the currents appropriately chosen, arbitrary
radiation patterns can be formed by coherent superposition. The interested
reader should refer to the electrical engineering literature for detailed
treatments of antenna arrays.
For the halfwave and fullwave antennas the angular distributions can
be integrated over angles to give
P = '
c
kd = tt
(9.58)
1 P'/l  cos A .
2 Jo \^) dt '
The integrals in (9.58) can be expressed in terms of the cosine integral,
Ci(x)=  f°°2°il^ (9.59)
J X t
f x / l  cos t \ df = ln ^ x) _ a ^ (9 6Q)
as follows :
280 Classical Electrodynamics
where y = 1.781 . . , is Euler's constant. Tables of the cosine integral
are given by Jahnke and Emde, pp. 69. The numerical results for the
power radiated are
72 (2.44, kd = tt
P =  (9.61)
2c (6.70, kd = 2tt
For a given peak current / the fullwave, centerfed antenna radiates
nearly 3 times as much power as the halfwave antenna. The coefficient
of P/2 has the dimensions of a resistance and is called the radiation
resistance R Tad of the antenna. The value in ohms is obtained from (9.61)
by multiplying the numbers by 30 (actually the multiplier is the numerical
value of the velocity of light divided by appropriate powers of 10). Thus
the half and fullwave centerfed antennas have radiation resistances of
73.2 ohms and 201 ohms, respectively.
The reader should be warned that the idealized problem of an infinitely
thin, linear antenna with a sinusoidal current distribution is a somewhat
simplified version of what occurs in practice. Finite lateral dimensions,
ohmic and radiative losses, nonsinusoidal current distributions, finite gaps
for excitation, etc., all introduce complications. These problems are
important in practical applications and are treated in detail in an extensive
literature on antenna design, to which the interested reader may refer.
9.5 Kirchhoff's Integral for Diffraction
The general problem of diffraction involves a wave incident on one or
more obstacles or apertures in absorbing or conducting surfaces. The
wave is scattered and perhaps absorbed, leading to radiation propagating
in directions other than the incident direction. The calculation of the
radiation emerging from a diffracting system is the aim of all diffraction
theories. The earliest systematic attempt was that of G. Kirchhoff (1882),
based on the ideas of superposition of elemental wavelets due to Huygens.
In this section we will discuss Kirchhoff's method and point out some of
its deficiencies, and in the next section derive vector theorems which
correspond to the basic scalar theorem of Kirchhoff.
The customary geometrical situation in diffraction is two spatial regions
I and II separated by a boundary surface S, as shown in Fig. 9.4. For
example, S may be an infinite metallic sheet with certain apertures in it.
The incident wave, generated by sources in region I, approaches S from
one side and is diffracted at the boundary surface, giving rise to scattered
waves, one transmitted and one reflected. It is usual to consider only the
transmitted wave and call its distribution in angle the diffraction pattern
[Sect. 9.5]
Simple Radiating Systems and Diffraction
281
E r , B r
E t , B(
Eo,Bo
■*■ E 0) Bo
Fig. 9.4 Diffracting system. The surface
S, with certain apertures in it, gives rise to
reflected and transmitted fields in regions I
and II in addition to the fields which would
be present in the absence of the surface.
of the system. If the incident wave is described by the fields E , B , the
reflected wave by the fields E r , B r , and the transmitted wave by E t , B t ,
then the total fields in regions I and II are E = E + E s , B = B + B s ,
where s stands for r or /. The basic problem is to determine (E t , B t ) and
(E r , B r ) from the incident fields (E , B ) and the properties of the boundary
surface S. To connect the fields in region I with those in region II
boundary conditions for E and B must be satisfied on S, the form of these
boundary conditions depending on the properties of S.
The method of attack used in solving such problems is the Green's
theorem technique, as applied to the wave equation in Chapter 6. Con
sider a scalar field ip(x, t) defined on and inside a closed surface S and
satisfying the sourcefree wave equation in that region. The field ^(x, t)
can be thought of as a rectangular component of E or B. We proved in
Chapter 6 that the value of ip inside S could be written in terms of the value
of ip and its normal derivative on the surface as
4tt JsR
R
V>(x', t')  £ <Kx', O  —
R ay>(x', Q"
n da'
R 2 cR dt' U
(9.62)
where R = x — x', n is the outwardly directed normal to the surface, and
ret means evaluated at a time t' = t — (R/c). If a harmonic time depen
dence e~ ia * is assumed, this integral form for ip(x, i) can be written:
ikR
w(x) = — <P n
4ttJs R
<t — n V>+ifc(l + — v da' (9.63)
Is R I \ kR/R J
To adapt (9.63) to diffraction problems we consider the closed surface
S to be made up of two surfaces S 1 and S 2 . Surface 5 X will be chosen as a
convenient one for the particular problem to be solved (e.g., the con
ducting screen with apertures in it), while surface S 2 will be taken as a
sphere or hemisphere of very large radius (tending to infinity) in region II,
as shown in Fig. 9.5. Since the fields in region 11 are the transmitted fields
282
Classical Electrodynamics
which originate from the diffracting region, they will be outgoing waves in
the neighborhood of S 2 . This means that the fields, and therefore ip(x),
will satisfy the radiation condition,
V/(M)
1 dtp
tp dr
'*;
(9.64)
With this condition on tp it can readily be seen that the integral in (9.63)
over the hemisphere S 2 vanishes inversely as the hemisphere radius as that
radius goes to infinity. Then we obtain the Kirchhoff integral for tp(x) in
region II :
y<x)
= _J_f £!!_„. v>+ ikll + — )tp da' (9.65)
4ttJ Si R i \ kR/R J
where n is now a unit vector normal to S x and pointing into region II.
In order to apply the Kirchhoff formula (9.65) to a diffraction problem
it is necessary to know the values of tp and dtp/dn on the surface S v Unless
we have already solved the problem exactly, these values are not known.
If, for example, S ± is a plane, perfectly conducting screen with an opening
in it and tp represents the component of electric field parallel to S v then we
know that tp vanishes everywhere on S x , except in the opening. But the
value of tp in the opening is undetermined. Without additional knowledge,
only approximate solutions can be found by making some assumption
about tp and dtp/dn on S v The Kirchhoff approximation consists of the
assumptions :
1 . tp and dtp/dn vanish everywhere on S t except in the openings.
2. The values of tp and dtp/dn in the openings are equal to the values of
the incident wave in the absence of any screens or obstacles.
The standard diffraction calculations of classical optics are all based on the
Kirchhoff approximation. It should be obvious that the recipe can have
only very approximate validity. There is a basic mathematical incon
sistency in the assumptions. It was shown for Laplace's equation (and
equally well for the Helmholtz wave equation) in Section 1.9 that the
Sources
Fig. 9.5 Possible diffraction
geometries. Region I contains
the sources of radiation. Region
II is the diffraction region, where
the fields satisfy the radiation
condition.
[Sect. 9.6] Simple Radiating Systems and Diffraction 283
solution inside a closed volume is determined uniquely by specifying ip
(Dirichlet boundary condition) or dip/dn (Neumann boundary condition)
on the surface. Both ip and dtp/dn cannot be given on the surface. The
Kirchhoff approximation works best in the shortwavelength limit in
which the diffracting openings have dimensions large compared to a wave
length. Being a scalar theory, even there it cannot account for details of
the polarization of the diffracted radiation. In the intermediate and
longwavelength limit, the scalar approximation fails badly, aside from the
drastic approximations inherent in the basic assumptions listed above.
Since the diffraction of electromagnetic radiation is a boundaryvalue
problem in vector fields, we expect that a considerable improvement can
be made by developing vector equivalents to the Kirchhoff integral (9.65).
9.6 Vector Equivalents of Kirchhoflf Integral
To obtain vector equivalents to the Kirchhoff integral (9.63) we first
note that with the definition,
G(x,x') =  — (9.66)
477 R
the scalar form (9.63) can be written
w (x) = i> [Gn • V>  yn • V'G] da' (9.67)
By writing down the result (9.67) for each rectangular component of the
electric or magnetic field and combining them vectorially, we can obtain
the vector theorem,
E(x) = <f> [G(n • V')E  E(n • V'G)] da' (9.68)
with a corresponding relation for B. This result is not a particularly
convenient one for calculations. It can be transformed into a more useful
form by a succession of vector manipulations. First the integrand in
(9.68) can be written
[ ] = (n • V')(GE)  2E(n • V'G) (9.69)
Then the vector identities,
n x (E x V'G) = E(n • V'G)  (n • E)V'G 1
V'G x (n x E) = n(E • V'G)  E(n • V'G) J
can be combined to eliminate the last term in (9.69) :
[ ] = (n . V')(GE)  n x (E x V'G)  n(E • V'G)
(n • E)V'G  (n x E) x V'G (9.71)
284 Classical Electrodynamics
Now the curl of the product of a vector and a scalar is used to transform
the second term in (9.71), while the fact that V • E = is used to re
express the third term. The result is
[ ] = ( n • V')(GE) + n x V x (GE)  nV • (GE)
(n • E)V'G  (n x E) x V'G  Gn x (V x E)
(9.72)
While it may not appear very fruitful to transform the two terms in (9.68)
into six terms, we will now show that the surface integral of the first three
terms in (9.72), involving the product (GE), vanishes identically. To do
this we make use of the following easily proved identities connecting
surface integrals over a closed surface S to volume integrals over the interior
of S:
i> A • n da = V • A d 3 x
Js Jv
(p(nxA)Ja= V x A d 3
Js Jv
\ <f>n da = V<f> cPx
Js Jv
(9.73)
where A and <f> are any wellbehaved vector and scalar functions. With
these identities the surface integral of the first three terms in (9.72) can be
written
<j> [(n • V')(GE) + n x V x (GE)  nV • (GE)] da'
= [V' 2 (GE) + V x V x (GE)  V'(V • (GE))] dV (9.74)
Jv
From the expansion, V x V x A = V(V • A) — V 2 A, it is evident that
the volume integral vanishes identically. *
With the surface integral of the first three terms in (9.72) identically
zero, the remaining three terms give an alternative form for the vector
Kirchhoff relation (9.68). From Maxwell's equations we have V x E =
ikB, so that the final result for the electric field anywhere inside the volume
* The reader may well be concerned that theorems (9.73) do not apply, since the
vector function (GE) is singular at the point x' = x. But if the singularity is excluded by
taking the surface S as an outer surface 5" and a small sphere S" around x' = x, the con
tribution of the integral over S" can be shown to vanish in the limit that the radius of
S" goes to zero. Hence result (9.74) is valid, even though G is singular inside the volume
of interest.
[Sect. 9.6] Simple Radiating Systems and Diffraction 285
bounded by the surface S is
E(x) =  <j> [i/c(n x B)G + (n x E) x V'G + (n • E)V'G] da' (9.75)
The analogous expression for the magnetic induction is
B(x) =  &> lik(n x E)G + (n x B) x V'G + (n • B)V'G] da' (9.76)
In (9.75) and (9.76) the unit vector n is the usual outwardly directed normal.
These integrals have an obvious interpretation in terms of equivalent
sources of charge and current. The normal component of E in (9.75) is
evidently an effective surfacecharge density. Similarly, according to
(8.14), the tangential component of magnetic induction (n x B) acts as an
effective surface current. The other terms (n • B) and (n x E) are effective
magnetic surface charge and current densities, respectively.
Vector formulas (9.75) and (9.76) serve as vector equivalents to the
HuygensKirchhoff scalar integral (9.63). If the fields E and B are assumed
to obey the radiation condition (9.64) with the added vectorial relationship,
E = B x (r/r), it is easy to show that the surface integral at infinity
vanishes. Then, in the notation of Fig. 9.5, the electric field (9.75) is
E(x) = [(n x E) x V'G + (n • E)V'G + ik(n x B)G] da' (9.77)
where S x is the surface appropriate to the diffracting system, and n is now
directed into the region of interest.
The vector theorem (9.77) is a considerable improvement over the
scalar expression (9.65) in that the vector nature of the electromagnetic
fields is fully included. But to calculate the diffracted fields it is still
necessary to know the values of E and B on the surface S v The Kirchhoff
approximations of the previous section can be applied in the shortwave
length limit. But the sudden discontinuity of E and B from the unperturbed
values in the "illuminated" region to zero in the "shadow" region on the
back side of the diffracting system must be compensated for mathemati
cally by line currents around the boundaries of the openings. *
A very convenient formula can be obtained from (9.77) for the special
case of plane boundary surface S v We imagine that the surface S ±
containing the sources in the righthand side of Fig. 9.5 is changed in
shape into a large, flat pancake, as shown in Fig. 9.6. The region II of
"transmitted" fields now becomes two regions, II and II', connected
together only by an annular opening at infinity. We denote the two sides
* For a discussion of these line currents, see Stratton, pp. 468470, and Silver,
Chapter 5.
286
Classical Electrodynamics
Fig. 9.6
of the disc by S x and S±. The unit vectors n and n' = — n are directed into
regions II and II', respectively. Our aim is to obtain an integral form for
the fields in region II in terms of the fields specified on the righthand
surface S x . This is analogous to the geometrical situation shown in the
left side of Fig. 9.5. We do not care about the values of the fields in region
II'. In fact, the hypothetical sources inside the disc will be imagined to be
such that the fields in region II' give a contribution to the surface integral
(9.77) which makes the final expression for the diffracted fields in region II
especially useful. Once we have obtained the desired result [equation
(9.82) below] for the fields in region II as an integral over the surface S x ,
we will forget about the manner of derivation and ignore the whole left
hand side of Fig. 9.6. Our interest is in the diffracted fields in region II
caused by apertures or obstacles located on the plane surface S v
If the fields in regions II and II' are E, B and E', B', respectively, then
from the figure it is evident that when the thickness of the disc becomes
vanishingly small, integral (9.77) may be written
E(x) = J [(n x (E  E')) x V'G + n • (E  E')V'G
+ ikn x (B  B')G] da' (9.78)
The field E(x) on the left side is either E or E', depending on where the
point x lies. But the integral is over the righthand surface S x only.
One of the most common applications is to conducting surfaces with
apertures in them. The boundary conditions at a perfectly conducting
surface are n x E = 0, n • B = 0, but n • E =£ 0, n x B ^ 0. In cal
culating the surface integral in (9.78) it would be desirable to integrate
only over the apertures in the surface rather than over all of it. The first
[Sect. 9.6] Simple Radiating Systems and Diffraction 287
term in (9.78) exists only in the apertures if the screen is perfectly con
ducting. Consequently we try to choose the fields in region II' so that the
other terms vanish everywhere on S v Evidently we must choose
(n • E%' = (n • E) Sl }
(9.79)
(n x B%; = (n x B) Sl J
Of course, the fields E', B' must satisfy Maxwell's equations and the
radiation condition in region II' if E, B satisfy them in region II. It is easy
to show that the required relationship, giving (9.79) on the surfaces, is
n x E'(x') = n x E(x)
n • E'(x') = n • E(x)
n x B'(x') = n x B(x) (9 * 80)
n • B'(x') = n • B(x)
where the point x' is the mirror image of x in the plane S v The fields at
mirrorimage points have the opposite (same) values of tangential and
outwardly directed normal components of electric field (magnetic
induction).
With conditions (9.80) in (9.78) we obtain the simple result for the field
E(x) in terms of an integral over the plane surface S x bounding region II,*
E(x) = 2 (n x E) x V'G da' (9.81)
where (n x E) is the tangential electric field on S lt n is a unit normal
directed into region II, and G is the Green's function (9.66). Since
V = —V when operating on G, (9.81) can be put in the alternate form,
E(x) = 2V x  n x E(x')G(x, x') da' (9.82)
For a diffraction system consisting of apertures in a perfectly conducting
plane screen the integral over S t may be confined to the apertures only.
Result (9.81) or (9.82) is exact if the correct tangential component of E
over the apertures is inserted. In practice, we must make some approxi
mation as to the form of the aperture field. But, for plane conducting
screens at least, only the tangential electric field need be approximated
and the boundary conditions on the screen are correctly satisfied [as can
be verified explicitly from (9.82)].
* This form for plane screens was first obtained by W. R. Smythe, Phys. Rev., 72,
1066 (1947), using an argument based on the fields due to a double current sheet filling
the apertures, rather than the present Green'stheorem technique.
288
Classical Electrodynamics
9.7 Babinet's Principle of Complementary Screens
Before discussing examples of diffraction we wish to establish a useful
relation called Babinet's principle. Babinet's principle relates the dif
fraction fields of one diffracting screen to those of the complementary
screen. We first discuss the principle in the scalar Kirchhoff approxi
mation. The diffracting screen is assumed to lie in some surface S which
divides space into regions I and II in the sense of Section 9.5. The screen
occupies all of the surface S except for certain apertures. The comple
mentary screen is that diffracting screen which is obtained by replacing
the apertures by screen and the screen by apertures. If the surface of the
original screen is S a and that of the complementary screen is S b , then
S a + $b = S, as shown schematically in Fig. 9.7.
If there are sources inside S (in region I) which give rise to a field y(x),
then in the absence of either screen the field ip(x) in region II is given by
the Kirchhoff integral (9.65) where the surface integral is over the entire
surface S. With the screen S a in position, the field y> a (x) in region II is
given in the Kirchhoff approximation by (9.65) with the source field y) in
the integrand and the surface integral only over S b (the apertures).
Similarly, for the complementary screen S b , the field ip b (x) is given in the
same approximation by a surface integral over S a . Evidently, then, we
have the following relation between the diffraction fields tp a and \p b :
Wa + Wb = W
(9.83)
This is Babinet's principle as usually formulated in optics. If ip represents
an incident plane wave, for example, Babinet's principle says that the
S a
Fig. 9.7 A diffraction screen S a and its
complementary diffraction screen S b 
[Sect. 9.7] Simple Radiating Systems and Diffraction 289
diffraction pattern away from the incident direction is the same for the
original screen and its complement.
The above formulation of Babinet's principle is unsatisfactory in two
aspects : it is a statement about scalar fields, and it is based on the Kirchhoff
approximation. The second deficiency can be remedied by defining the
complementary problem as not only involving complementary screens
but also involving complementary boundary conditions (Dirichlet versus
Neumann) for the scalar fields. But since we are interested in the electro
magnetic field, we will not pursue the scalar problem further.
A rigorous statement of Babinet's principle for electromagnetic fields
can be made for a thin, plane, perfectly conducting screen and its comple
ment. We start by considering certain fields E , B incident on the screen
with metallic surface S a (see Fig. 9.7) in otherwise empty space. The
presence of the screen gives rise to transmitted and reflected fields, as
shown in Fig. 9.4. These transmitted and reflected fields will be denoted
collectively as scattered fields, E s , B s , unless we need to be more specific. For
a perfectly conducting screen, the surface current K induced by the incident
fields must be such that at all points on the screen's surface S a , n x E g =
— n x E . For a thin, plane surface, the symmetry of the problem implies
that the tangential components of scattered magnetic field at the surface
must be equal and opposite, being given from (5.90) by
n x H t = — K = n x H r (9.84)
c
where n points into the transmitted region II. As a matter of fact, by the
same arguments that led from (9.79) to (9.80), it can be established that
at any point x in region II and its mirrorimage point x' in region I, the
scattered fields satisfy the symmetry conditions,
n x E r (x') = n x E ( (x)
n • E r (x') = n • E t (x)
n x B r (x') = — n x B 4 (x)
nB r (x') = nB t (x)
(9.85)
It will be noted that these relations differ from those in (9.80) by having
the signs of E r (x') and B r (x') reversed. As we see from the work of Smythe
(pp. cit., Section 9.6), the fields of (9.80) correspond to a double layer of
current. The present fields have the symmetries (9.85) appropriate to a
single, plane, current sheet radiating in both directions.
An integral expression for the scattered magnetic induction can now be
written down in terms of the surface current K. Since B is the curl of the
290 Classical Electrodynamics
vector potential, we have
B s = V x — I KG da' (9.86)
c Js a
where G is the Green's function (9.66), and the integration goes over the
metallic surface S a of the screen. If we substitute for K from (9.84), we
can write the magnetic induction in region II as
B t (x) = 2V x n x B 4 (x')G(x, x') da (9.87)
Js a
This result is identical with (9.82) except that
(1) the roles of E and B have been interchanged,
(2) the present integration is only over the body of the screen, whereas that
in (9.82) is only over the apertures,
(3) the total electric field appears in (9.82), whereas only the scattered
fields occur in (9.87).
The comparison of (9.87) with (9.82) forms the basis of Babinet's
principle. If we write down the result (9.82) for the complement of the
screen with metallic surface S a , we have
E'(x) = 2V x n x E'(x')G(x, x') da' (9.88)
'So
The integration is only over S a , since that is the aperture in the comple
mentary screen. The field E' in region II is the sum,
E' = E ' + E/ (9.89)
where E ' is the incident electric field of the complementary diffraction
problem, and E/ the corresponding transmitted or diffracted field.
Evidently the two expressions (9.87) and (9.88) turn into one another under
the transformation,
B,>±(E ' + E/) (9.90)
It is easy to show that the other fields transform at the same time according
E t ^T(B ' + B/) (9.91)
the sign difference arising from the fact that the fields must represent
outgoing radiation in both cases. Since we could have started with the
complementary screen initially, it is clear that (9.90) and (9.91) must hold
equally with the primed and unprimed quantities interchanged. Com
parison of the two sets of expressions shows that the incident fields of the
original and complementary diffraction problems must be related accord
mgt ° E '=B , B ' = E (9.92)
The complementary problem involves not only the complementary screen,
[Sect. 9.7] Simple Radiating Systems and Diffraction 291
Fig. 9.8 Equivalent radiators according to Babinet's principle.
but also a complementary set of incident fields with the roles of E and B
interchanged.
The statement of Babinet's principle is therefore as follows: a dif
fracting system consists of a source producing fields E , B incident on a
thin, plane, perfectly conducting screen with certain apertures in it. The
complementary diffracting system consists of a source producing fields
E ' = — B , B ' = E incident on the complementary screen. If the
transmitted (diffraction) fields on the opposite side of the screens from the
source are E ( , B t and E/, B/ for the diffracting system and its complement,
respectively, then they are related by
E< + B/ = — E = — B '
(9.93)
B« — E/ = — B = +E '
These are the vector analogs of the scalar relation (9.83).
If a plane wave is incident on the diffracting screen, Babinet's principle
states that, in directions other than the incident direction, the intensity of
the diffraction pattern of the screen and its complement will be the same,
the fields being related by
E* = — B/
B, = E/
(9.94)
The polarization of the wave incident on the complementary screen must,
of course, be rotated according to (9.92).
The rigorous vector formulation of Babinet's principle is very useful in
microwave problems. For example, consider a narrow slot cut in an
infinite, plane, conducting sheet and illuminated with fields that have the
magnetic induction along the slot and the electric field perpendicular to
it, as shown in Fig. 9.8. The radiation pattern from the slot will be the
same as that of a thin linear antenna with its driving electric field along the
antenna, as considered in Sections 9.2 and 9.4. The polarization of the
radiation will be opposite for the two systems. Elaboration of these ideas
makes it possible to design antenna arrays by cutting suitable slots in the
sides of wave guides.*
* See, for example, Silver, Chapter 9.
292 Classical Electrodynamics
9.8 Diffraction by a Circular Aperture
The subject of diffraction has been extensively studied since Kirchhoff 's
original work, both in optics, where the scalar theory based on (9.65)
generally suffices, and in microwave generation and transmission, where
more accurate solutions are needed. There exist specialized treatises
devoted entirely to the subject of diffraction and scattering. We will
content ourselves with a few examples to illustrate the use of the scalar
and vector theorems (9.65) and (9.82) and to compare the accuracy of the
approximation schemes.
Historically, diffraction patterns were classed as Fresnel diffraction and
Fraunhofer diffraction, depending on the distance of the observation point
from the diffracting system. Generally the diffracting system (e.g., an
aperture in an opaque screen) has dimensions comparable to, or large
compared to, a wavelength. Then the observation point may be in the
near zone, less than a wavelength away from the diffracting system. The
nearzone fields are complicated in structure and of little interest. Points
many wavelengths away from the diffracting system, but still near the
system in terms of its own dimensions, are said to lie in the Fresnel zone.
Further away, at distances large compared to both the dimensions of
the diffracting system and the wavelength, lies the Fraunhofer zone. The
Fraunhofer zone corresponds to the radiation zone of Section 9. 1 . The
diffraction patterns in the Fresnel and Fraunhofer zones show character
istic differences which come from the fact that for Fresnel diffraction the
region of the diffracting system nearest the observation point is of greatest
importance, whereas for Fraunhofer diffraction the whole diffracting
system contributes. We will consider only Fraunhofer diffraction, leaving
examples of Fresnel diffraction to the problems at the end of the chapter.
If the observation point is far from the diffracting system, expansion
(9.7) can be used for R = x — x'. Keeping only lowestorder terms in
(1/Ar), the scalar Kirchhoff expression (9.65) becomes
=  e —[
4irr Js
y(x) =  — \ e
477T J Si
tkx'
n • V'y(x') + ik • n^(x')
da' (9.95)
where x' is the coordinate of the element of surface area da', r is the length
of the vector x from the origin O to the observation point P, and k =
k(x/r) is the wave vector in the direction of observation, as indicated in
Fig. 9.9. For a plane surface the vector expression (9.82) reduces in this
limit to
ie ikr f t <
E(x) = — k x n x E(x>~ Jk ' x da' (9.96)
2irr JSi
[Sect. 9.8]
Simple Radiating Systems and Diffraction
293
Fig. 9.9
As an example of diffraction we consider a plane wave incident at an
angle a on a thin, perfectly conducting screen with a circular hole of radius
a in it. The polarization vector of the incident wave lies in the plane of
incidence. Figure 9.10 shows an appropriate system of coordinates. The
screen lies in the xy plane with the opening centered at the origin. The
wave is incident from below, so that the domain z > is the region of
diffraction fields. The plane of incidence is taken to be the xz plane. The
incident wave's electric field, written out explicitly in rectangular com
ponents, is , x ifc(cosaz + sin as) (Q Qn\
E; = E (e x cos a — € 3 sin a)e (9.97)
In calculating the diffraction field with (9.95) or (9.96) we will make the
customary approximation that the exact field in the surface integral may
be replaced by the incident field. For the vector relation (9.96) we need
(n x E^ =0 = £ e 2 cos xe ik sina *' (9.98)
Then, introducing plane polar coordinates for the integration over the
opening, we have
E(x) =
ie ikr E cos a
(k x € 2 )
pdp\ c
o Jo
dpe
t'fcpfsinacos/3— sin0cos(<£— /?)]
2t7T * ~J ' 'Jo ' (9.99)
where 6, <f> are the spherical angles of k. If we define the angular function,
I = (sin 2 + sin 2 a  2 sin 6 sin a cos <f>)* (9.100)
Fig. 9.10 Diffraction by a circu
lar hole of radius a.
294 Classical Electrodynamics
the angular integral can be transformed into
— \ dp = —\ d^e~ ap6coaP ' = J (k P £) (9.101)
Itt Jo 2tt Jo
Then the radial integral in (9.99) can be done directly. The resulting
electric field in the vector Kirchhoff approximation is
E(x) = a 2 E cos a(k x e 2 )
Ji(ka£)
r ka£
The timeaveraged diffracted power per unit solid angle is
— = p i cos a — — (cos 2 <f> + cos 2 6 sin 2 <f>)
dLl 4tt
2/ 1 (fca£)
where
/ca
P,
Tra cos a
(9.102)
(9.103)
(9.104)
is the total power normally incident on the aperture. If the opening is large
compared to a wavelength {ka > 1), the factor [2J x {kak)lkai\ 2 peaks
sharply to a value of unity at £ = and falls rapidly to zero (with small
secondary maxima) within a region A£ ~ (l/ka) on either side of £ = 0.
This means that the main part of the wave passes through the opening in
the manner of geometrical optics; only slight diffraction effects occur.
For ka ~ 1 the Besselfunction varies comparatively slowly in angle; the
transmitted wave is distributed in directions very different from the
incident direction. For ka < 1, the angular distribution is entirely deter
mined by the factor (k x e 2 ) in (9.102). But in this limit the assumption of
an unperturbed field in the aperture breaks down badly.
The total transmitted power can be obtained by integrating (9.103) over
all angles in the forward hemisphere. The ratio of transmitted power to
incident power is called the transmission coefficient T:
cos a C 2 * C" /2
T= cosa ^ (cos 2 <^ + cos 2 sin 2 0)
TV Jo Jo
J^kaZ)
sin Odd (9.105)
In the two extreme limits ka > 1 and ka < 1, the transmission coefficient
approaches the values,
f cos a, ka > 1
T^ I (9.106)
[ %{kdf cos a, ka < 1
The long wavelength limit {ka < 1) is suspect because of our approxi
mations, but it shows that the transmission is small for very small holes.
[Sect. 9.8] Simple Radiating Systems and Diffraction 295
For normal incidence (a = 0) the transmission coefficient (9.105) can be
written
'2
T=\ J t \ka sin 0)(
Jo \sm
With the help of the integral relations,
 sin dd
(9.107)
pr/2
Jo '
J n %z sin 0)
dd
sin
f«r/2
Jo '
J n \z sin 0)sin dd
Jo *
df
(9.108)
and the recurrence formulas (3.87) and (3.88), the transmission coefficient
can be put in the alternative forms,
T =
1 °°
1  — ^Jzm+iilka)
TO =
i /*2fco
1rf Ut)dt
2ka Jo
(9.109)
The transmission coefficient increases more or less monotonically as ka
increases, with small oscillations superposed. For ka > 1, the second form
in (9.109) can be used to obtain an asymptotic expression,
T~ 1 
1
1
2ka 2jir{kay
sin
( 2k "l)
+
(9.110)
which exhibits the small oscillations explicitly. These approximate expres
sions (9.109) and (9.110) for Tgive the general behavior as a function of
ka, but are not very accurate. Exact calculations, as well as more accurate
approximate ones, have been made for the circular opening. These are
compared with each other in the book by King and Wu (Fig. 41, p. 126).
The correct asymptotic expression does not contain the l/2ka term in
(9.110), and the coefficient of the term in (ka) 3A is twice as large.
We now wish to compare our results of the vector Kirchhoff approxi
mation with the usual scalar theory based on (9.95). For a wave not
normally incident the question immediately arises as to what to choose for
the scalar function y(x). Perhaps the most consistent assumption is to
take the magnitude of the electric or magnetic field. Then the diffracted
intensity is treated consistently as proportional to the absolute square of
(9.95). If a component of E or B is chosen for ip, we must then decide
whether to keep or throw away radial components of the diffracted field in
296
Classical Electrodynamics
calculating the diffracted power. Choosing the magnitude of E for tp,
we have, by straightforward calculation with (9.95),
ip(x) = —ik — a 2 E \
2l7 /cos a + cos 6 \ J x (kaEi)
2 / kaS
(9.111)
as the scalar equivalent of (9. 102). The power radiated per unit solid angle
in the scalar Kirchhoff approximation is
dP _ (kaf
— ~ P, v —  cos
dQ. 4tt
/cos a + cos 6\ 2
\ 2 cos a /
2Ji(fca^)
ka£
(9.112)
where P t is given by (9.104).
If we compare the vector Kirchhoff result (9.103) with (9.112), we see
similarities and differences. Both formulas contain the same "diffraction"
distribution factor [J 1 (kag)/kag] 2 and the same dependence on wave
number. But the scalar result has no azimuthal dependence (apart from
that contained in ), whereas the vector expression does. The azimuthal
variation comes from the polarization properties of the field, and must be
absent in a scalar approximation. For normal incidence (a = 0) and
ka > 1 the polarization dependence is unimportant. The diffraction is
(a)
/
\
/
\
1
\
1
\
I
'
)/'
\ ,
1/
\\
//
\\
//
\
/
R *
(b)
Fig. 9.11 Fraunhofer diffraction pattern for a circular opening one wavelength in
diameter in a thin, plane, conducting sheet. The plane wave is incident on the screen
at 45°. The solid curves are the vector Kirchhoff approximation, while the dotted curves
are the scalar approximation, (a) The intensity distribution in the plane of incidence
(E plane), (b) The intensity distribution (enlarged 2.5 times) perpendicular to the plane
of incidence (H plane).
[Sect. 9.9] Simple Radiating Systems and Diffraction 297
confined to very small angles in the forward direction. Then both scalar
and vector approximations reduce to the common expression,
dP ^ p (kaf
dQ. l IT
J x (ka sin 6)
ka sin 6
(9.113)
The vector and scalar Kirchhoff approximations are compared in Fig.
9.11 for the angle of incidence equal to 45° and for an aperture one wave
length in diameter (ka — n). The angular distribution is shown in the plane
of incidence (containing the electric field vector of the incident wave) and a
plane perpendicular to it. The solid (dotted) curve gives the vector (scalar)
approximation in each case. We see that for ka = tt there is a considerable
disagreement between the two approximations. There is reason to believe
that the vector Kirchhoff result is close to the correct one, even though the
approximation breaks down seriously for ka < 1. The vector approxi
mation and exact calculations for a rectangular opening yield results in
surprisingly good agreement, even down to ka ~ \*
9.9 Diffraction by Small Apertures
In the largeaperture or shortwavelength limit we have seen that a
reasonably good description of the diffracted fields is obtained by approxi
mating the tangential electric field in the aperture by its unperturbed
incident value. For longer wavelengths this approximation begins to fail.
When the apertures have dimensions small compared to a wavelength, an
entirely different approach is necessary. We will consider a thin, fiat,
perfectly conducting sheet with a small hole in it. The dimensions of the
hole are assumed to be very small compared to a wavelength of the electro
magnetic fields which are assumed to exist on one side of the sheet. The
problem is to calculate the diffracted fields on the other side of the sheet.
Since the sheet is assumed flat, the simple vector theorem (9.82) is appro
priate. Evidently the problem is solved if we can determine the electric
field in the plane of the hole.
As pointed out by Bethe (1942), the fields in the neighborhood of the
aperture can be treated by static or quasistatic methods. In the absence
of the aperture the electromagnetic fields near the conducting plane
consist of a normal electric field E and a tangential magnetic induction
B on one side, and no fields on the other. By "near the conducting plane,"
we mean at distances small compared to a wavelength. If a small hole is
* See J. A. Stratton and L. J. Chu, Phys. Rev., 56, 99 (1939), for a series of figures
comparing the vector Kirchhoff approximation with exact calculations by P. M. Morse
and P. J. Rubenstein, Phys. Rev., 54, 895 (1938).
298
Classical Electrodynamics
Eq
Fig. 9.12
now cut in the plane, the fields will be altered and will penetrate through
the hole to the other side. But far away from the hole (in terms of its
dimensions), although still "near the conducting plane," the fields will be
the same as if the hole were not there, namely, normal E and tangential
B . The electric field lines might appear as shown in Fig. 9.12. Since the
departures of the fields E and B from their unperturbed values E and B
occur only in a region with dimensions small compared to a wavelength,
the task of determining E or B near the aperture becomes a problem in
electrostatics or magnetostatics, apart from the overall sinusoidal time
dependence e~ imt . For the electric field, it is a standard potential problem
of knowing the "asymptotic" values of E on either side of the perfectly
conducting sheet which is an ftquipotential surface. Similarly for the
magnetic induction, B must be found to yield B and zero "asymptotically"
on either side of the sheet, with no normal component on the surface. Then
the electric field due to the time variation of B can be calculated and
combined with the "electrostatic" electric field to give the total electric
field near the opening.
For a circular opening of radius a small compared to a wavelength, for
example, the tangential electric field in the plane of the opening can be
shown to be .
E tan = E J. + — (n x B )Va 2  P 2 (9.114)
W« 2 — p* ""
where E = E • n is the magnitude of the normal electric field in the
absence of the hole, B is the tangential magnetic induction in the absence
of the hole, n is the unit vector normal to the surface and directed into the
diffraction region [as in (9.82)], and p is the radius vector in the plane
measured from the center of the opening. With this tangential field deter
mined in the static limit it is a straightforward matter to determine the
[Sect. 9.10] Simple Radiating Systems and Diffraction 299
diffracted fields and power from (9.82). The calculations for the circular
opening will be left to the problems at the end of the chapter (Problems
9.10 and 9.11).
9.10 Scattering by a Conducting Sphere in the
Short Wavelength Limit
Another type of problem which is essentially diffraction is the scattering
of waves by an obstacle. We will consider the scattering of a plane
electromagnetic wave by a perfectly conducting obstacle whose dimensions
are large compared to a wavelength. For a thin, flat obstacle, the tech
niques of Section 9.8, perhaps with Babinet's principle, can be used. But
for other obstacles we base the calculation on vector theorem (9.77) for
the scattered fields. If we consider only the fields in the radiation zone
(kr > 1), the integral (9.77) for the scattered field E s becomes
E s — — f [(n x E s ) x k + (n • E s )k  fc(n x B s )> ik * x ' da'
4rrir Jst
(9.115)
where k is the wave vector of the scattered wave, and S x is the surface of
the obstacle. It will be somewhat easier to calculate with the magnetic
induction B s = (k x E s )/fc:
B s ^ — k xj f(n x E s ) x 7 n x B,
— ikx'
v „ .. „ e~ l **da' (9.116)
4mr Jsi L k
In the absence of knowledge about the correct fields E s and B s on the
surface of the obstacle, we must make some approximations. If the wave
length is short compared to the dimensions of the obstacle, the surface
can be divided approximately into an illuminated region and a shadow
region. * The boundary between these regions is sharp only in the limit of
geometrical optics. The transition region can be shown to have a width of
the order of (2lkR) 1A R, where R is a typical radius of curvature of the
surface. Since R is of the order of magnitude of the dimensions of the
obstacle, the shortwavelength limit will approximately satisfy the geo
metrical condition. In the shadow region the scattered fields on the surface
must be very nearly equal and opposite to the incident fields. In the
illuminated region, the scattered tangential electric field and normal
magnetic induction must be equal and opposite to the corresponding
incident fields in order to satisfy the boundary conditions on the surface
* For a very similar treatment of the scattering of a scalar wave by a sphere, see Morse
and Feshbach, pp. 15511555.
300
Classical Electrodynamics
of the perfectly conducting obstacle. On the other hand, the tangential B s
and normal E s in the illuminated region will be approximately equal to
the incident values, just as for an infinite, flat, conducting sheet, to the
extent that the wavelength is small compared to the radius of curvature.
Thus we obtain the following approximate values for the scattered fields
on the surface of the obstacle :
Shadow Region
E s ~E;
B. ~ B,
Illuminated Region
n x E s = — n x E,
n • B s = n • B,
n x B s ~ n x B^
n • E ~ n • E,
where E i5 B^ are the fields of the incident wave. With these boundary
values the scattered magnetic induction (9.116) can be written as
ATcr
B a
Airir
k x (F sh + Fin)
where
'sh
JshL/c
x E,) + n x B,
— ikx'
da'
is the integral over the shadow region, and
Fin = I
Jil
x(nxEi)nxBi
k
ikx'
da
(9.117)
(9.118)
(9.119)
is the integral over the illuminated region.
If the incident wave is a plane wave with wave vector k^
E/x) = E e ik °* \
B,(x) = ^° x E,(x) j
(9.120)
the integrals over the shadow and illuminated regions of the obstacle's
surface are
F sh = J f [(k + ko) x (n x E ) + (n . E )k o y' (k ° k)  x 'rfa'
Fill = r I [(k  ko) x (n x E )  (n . E )k ]e , " (k ° k)  1 ' da'
k Jill
(9.121)
These integrals behave very differently as functions of the scattering angle.
In the shortwavelength limit the magnitudes of k • x' and k • x' are large
compared to unity. Thus the exponential factors in (9.121) will oscillate
[Sect. 9.10] Simple Radiating Systems and Diffraction 301
rapidly and cause the integrands to have very small average values except
in the forward direction where k ^ 1^. In that direction the second term
in both F sh and F m is unimportant, since the scattered field (9.117) is
proportional to k x F. The behavior of the two contributions is thus
governed by the first terms in (9.121), at least in the forward direction. We
see that F sh and F m are proportional to (k ± 1%), respectively; the
shadow integral will be large and the integral from the illuminated region
will go to zero. As the scattering angle departs from the forward direction
the shadow integral will vanish rapidly, both the exponential and the
vector factor in the integrand having the same tendency. On the other
hand, the integral from the illuminated region will be small in the forward
direction and can be expected to be small at all angles, the exponential and
the vector factor in the integrand having opposite tendencies. The shadow
integral is evidently the diffraction contribution, while the integral from
the illuminated region is the reflected wave.
To proceed much further we must specify the shape of the obstacle. We
will assume that it is a perfectly conducting sphere of radius a. Since the
shadow integral is large only in the forward direction, we will evaluate it
approximately by placing k = k^, everywhere except in the exponential.
Then, omitting the second term in (9.121) and using spherical coordinates
on the surface of the sphere, we obtain
J'ff/2 f2ir
„•„ a „~„ iA"a(l— cos 0)cosa I in 
sin a d<x cos cue dp e
o Jo
, . r^. _ ,„ „ . c,„ „ „~ ™c ~ „ • . z'fca sine sin a cos (/?(£)
(9.122)
The angles 6, </> and a, /3 are those of k and n relative to 1^. The exponential
factor involving (1 — cos 6) can be set equal to unity, since at small angles
its exponent is a factor 0/2 smaller than the other exponent. The integral
over /? is 2rrJ (ka sin 6 sin a). Hence
"sh ~ —
r*/2
47ra 2 E I J (kad sin a)cos a sin a dot. (9.123)
Jo
where we have approximated sin 6 ~ 6. The integral over a is pro
fkaO
portional to the integral xJ Q {x) dx = kad J^kad). Therefore the shadow
cr'Cttte^rincr intorrrol if •'0
scattering integral is
F sh 47r 2 aE ^^ (9.124)
kad
We see that this is essentially the diffraction field of a circular aperture
(9.102).
The integral over the illuminated region, giving the reflected or back
scattered wave, is somewhat harder to evaluate. We must consider
302 Classical Electrodynamics
arbitrary scattering angles, since there is no enhancement in the forward
direction. Then the integral consists of a relatively slowly varying vector
function of angles times a rapidly varying exponential. As is well known,
the dominant contribution to such an integral comes from the region of
integration where the phase of the exponential is stationary. The phase
factor is
/(a, /?) = (k„ — k) • x' = ka[(l — cos 0) cos a — sin 6 sin a cos (/? — </>)]
(9.125)
The stationary point is easily shown to be at angles Oq, /? , where
77 0)
2 2 (9.126)
These angles are evidently just those appropriate for reflection from the
sphere according to geometrical optics. At this point the unit vector n
points in the direction of (k — Ilq). If we expand the phase factor around
a = a , /? = ($ , we obtain
1   (x 2 + cos 2  y 2 ) + • •
/(a, {$) = —2ka sin 
2 \ 2
(9.127)
where x = a — olq, y = /S — /9 . Then integral (9.121) can be approxi
mated by evaluating the square bracket at a = a , /? = /? :
F„i ^ a 2 sin 0[2(n o • E ) n  E > 2itosin W2)
x \dx e i[ka sin m)]x2 \dy e i[ka sin (e/2) cos2 (e/2)]2/2 (9 128)
where n,, is a unit vector in the direction (k — k ). Provided 6 is not too
small, the phase factors oscillate rapidly for large x or y. Hence the
integration can be extended to ± oo in each integral without error. Using
the result,
we obtain
J e Ux * dx= () e iw/ * (9.129)
Fiu ^ i ^ e~ 2ikasin (fl/2) [2(n • EoK  E ] (9.130)
k
After some vector algebra the contribution to the scattered field from the
illuminated part of the sphere can be written
,ifcr
E ( s iU) ^   £ — e~ 2ika sin W 2 > e m (9.131)
2 r
[Sect. 9.10]
Simple Radiating Systems and Diffraction
303
Fig. 9.13 Polarization of reflected
wave relative to the incident polari
zation.
where the polarization vector e m has a direction denned in Fig. 9.13. If
the polarization vector of the incident wave E makes an angle d with the
normal to the plane containing the wave vectors k and k„, the azimuthal
angle y of e m , measured from the plane containing k and ko, is given by
y _ (y/2) _ <5. We note that the reflected field (9.131) is constant in
magnitude as a function of angle, although it has a rapidly varying
phase.
The scattered electric field due to the shadow region is, from (9.124) and
(9.117),
E (sh) ^ ika 2 JJM1 g!!( k x e ) x k (9 132)
s had r k 2
Comparison of the two contributions to the scattered wave shows that in
the forward direction the shadow field is larger by a factor ka > 1. But
for angles much larger than ~ (l/ka) the shadow field becomes very
small and the isotropic reflected field dominates. The power scattered per
unit solid angle can be expressed in the form:
dP*
dQ
~ PA
(kaf
Att
_1_
47T '
U^kad)
kaO
<
ka
(9.133)
ka
where P t = (c£ 2 a 2 /8) is the incident power per unit area times the pro
jected area (rra 2 ) of the sphere. At small angles the scattering is a typical
diffraction pattern [see (9.1 13)]. At large angles the scattering is isotropic.
At intermediate angles the two amplitudes interfere, causing the scattered
power to have sharp minimum values considerably smaller than the
304
Classical Electrodynamics
Fig. 9.14 Diffraction pattern for
a conducting sphere, snowing the
forward peak due to shadow
scattering, the isotropic reflected
contribution, and the interference
maxima and minima.
isotropic value at certain angles, as shown in Fig. 9.14. The amount of
interference depends on the orientation of the incident polarization vector
relative to the plane of observation containing k and k . For E in this
plane the interference is much greater than for E perpendicular to it.*
The total power scattered is obtained by integrating over all angles.
Neglecting the interference terms, the total scattered power is the sum of
the integrals of the diffraction peak and the isotropic reflected part. The
integrals are easily shown to be equal in magnitude. Hence
P g = P, + i>< = IP,
(9.134)
We sometimes rephase this result by saying that the effective area of the
sphere for scattering (its scattering cross section) is IttcP. One factor of
77a 2 comes from the direct reflection; the other comes from the diffraction
scattering which must accompany the formation of a shadow behind the
obstacle.
Scattering of electromagnetic waves by a conducting sphere is treated
by another method, especially in the longwavelength limit, in Section 16.9.
REFERENCES AND SUGGESTED READING
The simple theory of radiation from a localized source distribution is discussed in all
modern textbooks. Treatments analogous to that given here may be found in
Panofsky and Phillips, Chapter 13,
Stratton, Chapter VIII.
* See King and Wu, Appendix, for numerous graphs of scattering by spheres as a
function of ka.
[Probs. 9] Simple Radiating Systems and Diffraction 305
More complete discussions of antennas and antenna arrays are given in engineering
works, such as
Jordan,
Kraus,
Schelkunoff,
Silver.
The subject of diffraction has a very extensive literature. A comprehensive treatment
of both the scalar Kirchhoff and the vector theory, with many examples and excellent
figures, is given by
Born and Wolf, Chapters VIII, IX, and XI.
A more elementary discussion of the scalar theory is found in
Slater and Frank, Chapters XIII and XIV.
Mathematical techniques for diffraction problems are discussed by
Baker and Copson,
Morse and Feshbach, Chapter 11.
The vector theorems used in Sections 9.69.10 are presented by
Morse and Feshbach, Chapter 13,
Silver, Chapter 5,
Stratton, Sections 8.14 and 8.15.
A specialized monograph on the scattering of electromagnetic waves by obstacles, with
an emphasis on useful numerical results, is the book by
King and Wu.
PROBLEMS
9.1 Discuss the power flow and energy content of the complete electric dipole
fields (9.18) in terms of the complex Poynting's vector S = (c/8tt)(E x B*)
and the timeaveraged energy density u = (1/16tt)(E • E* + B • B*). The
real part of S gives the true, resistive power flow, while the imaginary part
represents circulating, reactive power.
(a) Show that the real part of S is in the radial direction and is given by
r~~ 2 times equation (9.23).
(b) Show that the imaginary part of S has components in the r and
directions given by
ImS r = ^p 2 sin 2
lmS =  ^ 2 (1 + A: 2 / 2 ) sin B cos 6
Make a sketch to show the direction of circulating power flow by suitably
oriented arrows, the length of each arrow being proportional to the
magnitude of Im S at that point.
(c) Calculate the timeaveraged energy density:
1 3n(np) p 2 /c 2 n • p a A: 4 [n x p] 2
" ~ 16tt r 6 + 4nr* + Sirr*
(d) Derive Poynting's theorem for the complex Poynting's vector. To
what is Im (V • S) equal? Verify that this holds true for the results of (b)
and (c).
306 Classical Electrodynamics
9.2 A radiating quadrupole consists of a square of side a with charges ±q at
alternate corners. The square rotates with angular velocity co about an
axis normal to the plane of the square and through its center. Calculate
the quadrupole moments, the radiation fields, the angular distribution of
radiation, and the total radiated power in the longwavelength approxi
mation.
9.3 Two halves of a spherical metallic shell of radius R and infinite conductivity
are separated by a very small insulating gap. An alternating potential is
applied between the two halves of the sphere so that the potentials are
±V cos at. In the longwavelength limit, find the radiation fields, the
angular distribution of radiated power, and the total radiated power from
the sphere.
9.4 A thin linear antenna of length d is excited in such a way that the sinusoidal
current makes a full wavelength of oscillation as shown in the figure.
(a) Calculate exactly the power radiated per unit solid angle and plot
the angular distribution of radiation.
(b) Determine the total power radiated and find a numerical value for
the radiation resistance.
9.5 Treat the linear antenna of Problem 9.4 by the longwavelength multipole
expansion method.
(a) Calculate the multipole moments (electric dipole, magnetic dipole,
and electric quadrupole).
(b) Compare the angular distribution for the lowest nonvanishing
multipole with the exact distribution of Problem 9.4.
(c) Determine the total power radiated for the lowest multipole and the
corresponding radiation resistance.
9.6 A perfectly conducting flat screen occupies onehalf of the xy plane
(i.e., x < 0). A plane wave of intensity I and wave number k is incident
along the z axis from the region z < 0. Discuss the values of the diffracted
fields in the plane parallel to the xy plane defined by z — Z > 0. Let the
coordinates of the observation point be (X, 0, Z).
(a) Show that, for the usual scalar Kirchhoff approximation and in the
limit Z > X, the diffracted field is
Y(X, 0, Z) =* i^gikziwtl 1 + A jl ("%«"«/,
where £ = (k/2Z)^X.
(b) Show that the intensity can be written
/ =
&(c(4) + i) 2 + m) + m
where C(f) and S(€) are the socalled Fresnel integrals. Determine the
asymptotic behavior of I for £ large and positive (illuminated region) and I
[Probs. 9] Simple Radiating Systems and Diffraction 307
large and negative (shadow region). What is the value of / at X = ?
Make a sketch of / as a function of X for fixed Z.
(c) Use the vector formula (9.82) to obtain a result equivalent to that of
part (a). Compare the two expressions.
9.7 A linearly polarized plane wave of amplitude E and wave number k is
incident on a circular opening of radius a in an otherwise perfectly con
ducting flat screen. The incident wave vector makes an angle a with the
normal to the screen. The polarization vector is perpendicular to the plane
of incidence.
{a) Calculate the diffracted fields and the power per unit solid angle
transmitted through the opening, using the vector Kirchhoff formula
(9.82) with the assumption that the tangential electric field in the opening
is the unperturbed incident field.
(b) Compare your result in part (a) with the standard scalar Kirchhoff
approximation and with the result in Section 9.8 for the polarization
vector in the plane of incidence.
9.8 A rectangular opening with sides of length a and b > a defined by
x = ±(a/2), y = ±(6/2) exists in a flat, perfectly conducting plane sheet
filling the xy plane. A plane wave is normally incident with its polarization
vector, making an angle yS with the long edges of the opening.
{a) Calculate the diffracted fields and power per unit solid angle with the
vector Kirchhoff relation (9.82), assuming that the tangential electric field
in the opening is the incident unperturbed field.
(b) Calculate the corresponding result of the scalar Kirchhoff approxi
mation.
(c) For b = a, P = 45°, ka = 4n, compute the vector and scalar approxi
mations to the diffracted power per unit solid angle as a function of the
angle 6 for <f> = 0. Plot a graph showing a comparison between the two
results.
9.9 A cylindrical coaxial transmission line of inner radius a and outer radius b
has its axis along the negative z axis. Both inner and outer conductors end
at z = 0, and the outer one is connected to an infinite plane flange occupy
ing the whole xy plane (except for the annulus of radius b around the
origin). The transmission line is excited at frequency o> in its dominant
TEM mode, with the peak voltage between the cylinders being V. Use the
vector Kirchhoff approximation to discuss the radiated fields, the angular
distribution of radiation, and the total power radiated.
9.10 Discuss the diffraction due to a small, circular hole of radius a in a flat,
perfectly conducting sheet, assuming that ka < 1.
(a) If the fields near the screen on the incident side are normal E e ltof
and tangential B e iait , show that the diffracted electric field in the
Fraunhofer zone is
a ikr —ieot
E = k 2 a 3
lirr
2*x
k
b ° + H e ° x 9"
where k is the wave vector in the direction of observation.
(b) Determine the angular distribution of the diffracted radiation and
show that the total power transmitted through the hole is
308 Classical Electrodynamics
9.11 Specialize the discussion of Problem 9.10 to the diffraction of a plane wave
by the small, circular hole. Treat the general case of oblique incidence at an
angle a to the normal, with polarization in and perpendicular to the plane
of incidence.
(a) Calculate the angular distributions of the diffracted radiation and
compare them to the vector Kirchhoff approximation results of Section 9.8
and Problem 9.7 in the limit ka < 1 .
(b) Show that the transmission coefficients [defined above (9.105)] for
the two states of polarization are
T„ =
64
27tt 2
(W /4+sin*a\
\ 4 COS a /
64
Th 2
T L = ^2 (kaf cos a
Note that these transmission coefficients are a factor (ka) 2 smaller than
those given by the vector Kirchhoff approximation in the same limit.
10
Magnetohydrodynamics
and Plasma Physics
10.1 Introduction and Definitions
Magnetohydrodynamics and plasma physics both deal with the
behavior of the combined system of electromagnetic fields and a con
ducting liquid or gas. Conduction occurs when there are free or quasifree
electrons which can move under the action of applied fields. In a solid
conductor, the electrons are actually bound, but can move considerable
distances on the atomic scale within the crystal lattice before making
collisions. Dynamical effects such as conduction and Hall effect are
observed when fields are applied to the solid conductor, but mass motion
does not in general occur. The effects of the applied fields on the atoms
themselves are taken up as stresses in the lattice structure. For a fluid, on
the other hand, the fields act on both electrons and ionized atoms to
produce dynamical effects, including bulk motion of the medium itself.
This mass motion in turn produces modifications in the electromagnetic
fields. Consequently we must deal with a complicated coupled system of
matter and fields.
The distinction between magnetohydrodynamics and the physics of
plasmas is not a sharp one. Nevertheless there are clearly separated
domains in which the ideas and concepts of only one or the other are
applicable. One way of seeing the distinction is to look at the way in which
the relation J = oE is established for a conducting substance. In the
simple model of Section 7.8 the electrons are imagined to be accelerated by
the applied fields, but to be altered in direction by collisions, so that their
motion in the direction of the field is opposed by an effective frictional
force vm\, where v is the collision frequency. Ohm's law just represents a
309
310 Classical Electrodynamics
balance between the applied force and the frictional drag. When the
frequency of the applied fields is comparable to v, the electrons have time
to accelerate and decelerate between collisions. Then inertial effects enter
and the conductivity becomes complex. Unfortunately at these same
frequencies the description of collisions in terms of a frictional force tends
to lose its validity. The whole process becomes more complicated. At
frequencies well above the collision frequency another thing happens. The
electrons and ions are accelerated in opposite directions by electric fields
and tend to separate. Strong electrostatic restoring forces are set up by
this charge separation. Oscillations occur in the charge density. These
highfrequency oscillations are called plasma oscillations and are to be
distinguished from lowerfrequency oscillations which involve motion of
the fluid, but no charge separation. These lowfrequency oscillations are
called magnetohydrodynamic waves.
In conducting liquids or dense ionized gases the collision frequency is
sufficiently high even for very good conductors that there is a wide
frequency range where Ohm's law in its simple form is valid. Under the
action of applied fields the electrons and ions move in such a way that,
apart from a highfrequency jitter, there is no separation of charge.
Electric fields arise from motion of the fluid which causes a current flow,
or as a result of timevarying magnetic fields or charge distributions
external to the fluid. The mechanical motion of the system can then be
described in terms of a single conducting fluid with the usual hydro
dynamic variables of density, velocity, and pressure. At low frequencies
it is customary to neglect the displacement current in Ampere's law. This
is then the approximation which is called magnetohydrodynamics.
In less dense ionized gases the collision frequency is smaller. There
may still be a lowfrequency domain where the magnetohydrodynamic
equations are applicable to quasistationary processes. Frequently astro
physical applications fall in this category. At higher frequencies, however,
the neglect of charge separation and of the displacement current is not allow
able. The separate inertial effects of the electrons and ions must be included
in the description of the motion. This is the domain which we call plasma
physics. There is here a range of physical conditions where a twofluid
model of electrons and ions gives an approximately correct description of
various phenomena. But for high temperatures and low densities, the
finite velocity spreads of the particles about their mean values must be
included. Then the description is made in terms of the Boltzmann
equation, with or without shortrange correlations. We will not attempt
to go into such details here. At still higher temperatures and lower
densities, the electrostatic restoring forces become so weak that the length
scale of charge separation becomes large compared to the size of the
[Sect. 10.2] Magnetohydrodynamics and Plasma Physics 311
volume being considered. Then the collective behavior implicit in a fluid
model is gone completely. We have left a few rapidly moving charged
particles interacting via Coulomb collisions. A plasma is, by definition,
an ionized gas in which the length which divides the smallscale individual
particle behavior from the largescale collective behavior is small com
pared to the characteristic lengths of interest. This length, called the
Debye screening radius, will be discussed in Section 10. 10. It is numerically
equal to 7.91 (T/n) 1A cm, where T is the absolute temperature in degrees
Kelvin and n is the number of electrons per cubic centimeter. For all but
the hottest or most tenuous plasmas it is small compared to 1 cm.
10.2 Magnetohydrodynamic Equations
We first consider the behavior of an electrically neutral, conducting
fluid in electromagnetic fields. For simplicity, we assume the fluid to be
nonpermeable. It is described by a matter density p(x, t), a velocity v(x, /),
a pressure p(x, t) (taken to be a scalar), and a real conductivity a. The
hydrodynamic equations are the continuity equation
^ + V.( P v) = (10.1)
ot
and the force equation :
/>7=Vp + (JxB) + F„+ P g (10.2)
dt c
In addition to the pressure and magneticforce terms we have included
viscous and gravitational forces. For an incompressible fluid the viscous
force can be written
F v = r)V 2 v (10.3)
where r\ is the coefficient of viscosity. It should be emphasized that the
time derivative of the velocity on the left side of (10.2) is the convective
derivative,
l = l + y.V (10.4)
dt dt
which gives the total time rate of change of a quantity moving instanta
neously with the velocity v.
312 Classical Electrodynamics
With the neglect of the displacement current, the electromagnetic fields
in the fluid are described by
VxE + i^O 1
c dt
VxB = ^J
(10.5)
The condition V • J = 0, equivalent to the neglect of displacement
currents, follows from the second equation in (10.5). The two divergence
equations have been omitted in (10.5). It follows from Faraday's law that
(d/dt) V • B = 0, and the requirement V • B = can be imposed as an
initial condition. With the neglect of the displacement current, it is
appropriate to ignore Coulomb's law as well. The reason is that the
electric field is completely determined by the curl equations and Ohm's
law (see below). If the displacement current is retained in Ampere's law
and V • E = 4Trp e is taken into account, corrections of only the order of
(v 2 /c 2 ) result. For normal magnetohydrodynamic problems these are
completely negligible.
To complete the specification of dynamical equations we must specify
the relation between the current density J and the fields E and B. For a
simple conducting medium of conductivity a, Ohm's law applies, and the
current density is
J' = dE! (10.6)
where J' and E' are measured in the rest frame of the medium. For a
medium moving with velocity v relative to the laboratory, we must trans
form both the current density and the electric field appropriately. The
transformation of the field is given by equation (6.10). The current density
in the laboratory is evidently
J = J' + Pe v (10.7)
where p e is the electrical charge density. For a onecomponent conducting
fluid, p e = 0. Consequently, Ohm's law assumes the form,
J=(t(e + xBJ (10.8)
Sometimes it is possible to assume that the conductivity of the fluid is
effectively infinite. Then under the action of fields E and B the fluid flows
in such a way that
E +  (v x B) = (10.9)
c
is satisfied.
[Sect. 10.3] Magnetohydrodynamics and Plasma Physics 313
Equations (10.1), (10.2), (10.5), and (10.8), supplemented by an equation
of state for the fluid, form the equations of magnetohydrodynamics. In
the next section we will consider some of the simpler aspects of them and
will elaborate the basic concepts involved.
10.3 Magnetic Diffusion, Viscosity, and Pressure
The behavior of a fluid in the presence of electromagnetic fields is
governed to a large extent by the magnitude of the conductivity. The
effects are both electromagnetic and mechanical. We first consider the
electromagnetic effects. We will see that, depending on the conductivity,
quite different behaviors of the fields occur. The time dependence of the
magnetic field can be written, using (10.8) to eliminate E, in the form:
 = Vx(yxB)V 2 B (10.10)
dt Attg
Here it is assumed that a is constant in space. For a fluid at rest (10.10)
reduces to the diffusion equation
?5 = _£!_V 2 B (10.11)
dt Attg
This means that an initial configuration of magnetic field will decay away
in a diffusion time
r = *=£ (10.12)
where L is a length characteristic of the spatial variation of B. The time t
is of the order of 1 sec for a copper sphere of 1 cm radius, of the order of 10 4
years for the molten core of the earth, and of the order of 10 10 years for a
typical magnetic field in the sun.
For times short compared to the diffusion time t (or, in other words,
when the conductivity is so large that the second term in (10.10) can be
neglected) the temporal behavior of the magnetic field is given by
3D
u — = V x (v x B) (10.13)
dt
From (6.5) it can be shown that this is equivalent to the statement that the
magnetic flux through any loop moving with the local fluid velocity is
constant in time. We say that the lines of force are frozen into the fluid
and are carried along with it. Since the conductivity is effectively infinite,
314 Classical Electrodynamics
the velocity w of the lines of force (defined to be perpendicular to B) is
given by (10.9):
w = c (ExB) M)
B 2
This socalled "E X B drift" of both fluid and lines of force can be under
stood in terms of individual particle orbits of the electrons and ions in
crossed electric and magnetic fields (see Section 12.8).
A useful parameter to distinguish between situations in which diffusion
of the field lines relative to the fluid occurs and those in which the lines of
force are frozen in is the magnetic Reynolds number R M . If V is a velocity
typical of the problem and L is a corresponding length, then the magnetic
Reynolds number is defined as
R M = ^ (10.15)
where r is the diffusion time (10.12). Transport of the lines of force with
the fluid dominates over diffusion if R M > 1 . For liquids like mercury or
sodium in the laboratory R M < 1, except for very high velocities. But in
geophysical and astrophysical applications R M can be very large compared
to unity.
The mechanical behavior of the system can be studied with the force
equation (10.2). Substituting for J from (10.8), we find
P7 = F4 2 (vi w) (10.16)
dt c 2
where F is the sum of all the nonelectromagnetic forces, and y is the
component of velocity perpendicular to B. From (10.16) it is apparent
that flow parallel to B is governed by the nonelectromagnetic forces alone.
The velocity of flow of the fluid perpendicular to B, on the other hand,
decays from some initially arbitrary value in a time of the order of
pc 2
T =
—. ; (1017)
oB 2
to a value
v i = W + ^ i F 1 (10.18)
CfB
In the limit of infinite conductivity this result reduces to that of (10.14),
as expected. The term proportional to B 2 in (10.16) is an effective viscous
or frictional force which tends to prevent flow of the fluid perpendicular to
the lines of magnetic force. Sometimes it is described as a magnetic
viscosity. If ordinary viscosity, here lumped into F, is comparable to the
[Sect. 10.3] Magnetohydrodynamics and Plasma Physics 315
magnetic viscosity, then the decay time r is shortened by an obvious factor
involving the ratio of the two viscosities.
The above considerations have shown that if the conductivity is large
the lines of force are frozen into the fluid and move along with it. Any
departure from that state decays rapidly away. In considering the
mechanical or electromagnetic effects we treated the opposite quantities as
given, but the equations are, of course, coupled. In the limit of very large
conductivity it is convenient to relate the current density J in the force
equation to the magnetic induction B via Ampere's law and to use the
infinite conductivity expression (10.9) to eliminate E from Faraday's law
to yield (10.13). The magnetic force term in (10.2) can now be written
 (J x B) =  — B x (V x B) (10.19)
C 477
With the vector identity
V(B • B) = (B • V)B + B x (V x B) (10.20)
Equation (10.19) can be transformed into
 (J x B) = V (f) + 1 (B • V)B (10.21)
This equation shows that the magnetic force is equivalent to a magnetic
hydrostatic pressure, 2
Pm = ^ (1022)
07T
plus a term which can be thought of as an additional tension along the
lines of force. The result (10.21) can also be derived frorh the Maxwell
stress tensor (see Section 6.9).
If we neglect viscous effects and assume that the gravitational force is
derivable from a potential g = — Vip, the force equation (10.2) takes the form
/»7=  V(p + p M + PV>) + 7" (B • V)B (10.23)
dt 4tt
In some simple geometrical situations, such as B having only one com
ponent, the additional tension vanishes. Then the static properties of the
fluid are described by „«.»
J p + p M + py> = constant (10.24)
This shows that, apart from gravitational effects, any change in mechanical
pressure must be balanced by an opposite change in magnetic pressure. If
the fluid is to be confined within a certain region so that p falls rapidly to
zero outside that region, the magnetic pressure must rise equally rapidly
in order to confine the fluid. This is the principle of the pinch effect
discussed in Section 10.5.
316
Classical Electrodynamics
10.4 Magnetohydrodynamic Flow between Boundaries with Crossed
Electric and Magnetic Fields
To illustrate the competition between freezing in of lines of force and
diffusion through them and between the E x B drift and behavior imposed
by boundary conditions, we consider the simple example of an incom
pressible, but viscous, conducting fluid flowing in the x direction between
two nonconducting boundary surfaces at z = and z = a, as shown in
Fig. 10.1. The surfaces move with velocities V x and V 2 , respectively, in
the x direction. A uniform magnetic field B acts in the zdirection. The
system is infinite in the x and y directions. We will look for a steadystate
solution for flow in the x direction in which the various quantities depend
only upon z.
If the fields do not vary in time, it is clear from Maxwell's equations
(10.5) that any electric field present must be an electrostatic field derivable
from a potential and determined solely by the boundary conditions, i.e.
an arbitrary external field. Expression (10.14) for the velocity of the lines
of force when a is infinite implies that there is an electric field in the y
direction. If we assume that to be the only component of E, then it must
be a constant, E . Because the moving fluid will tend to carry the lines of
force with it, we expect an x component BJz) of magnetic induction, as
well as the z component B .
The continuity equation (10. 1) reduces to V • v = for an incompressible
fluid. This is satisfied identically by a velocity in the x direction which
depends only on z. The force equation, neglecting gravity, has the steady
state form:
Vp = (J x B) + rjV 2 v
c
(10.25)
Fig. 10.1 Flow of viscous con
ducting fluid in a magnetic field
between two plane surfaces
moving with different velocities.
[Sect. 10.4] Magnetohydrodynamics and Plasma Physics
The only component of J that is nonvanishing is J y (z) :
J y (z) = a
E  B v(z)
c
317
(10.26)
where v is the x component of velocity. When we write out the three
component equations in (10.25), we find
dp aBJ^ B \ d 2 v
ox c \ c I oz
dp
dy
^ =
dp
dz
*(*.*.)
(10.27)
The magnetic force in the z direction is just balanced by the pressure
gradient. If we assume no pressure gradient in the x direction, the first of
these equations can be written :
d 2 v
dz 2
where
Tif— m
2„2\V£
M
\ TjC 2 I
(10.29)
is called the Hartmann number. From (10.17) M 2 can be seen to be the
ratio of magnetic to normal viscosity. The solution to (10.28), subject to
the boundary conditions v(0) = V x and v(a) = V 2 , is readily found to be
v(z) = 
V x
sinh M
sinh
In the limit B n
(10.30)
sinh M
0, M — *■ 0, we obtain the standard laminarflow result
z
v(z)= V X + {V %
a
V,)
(10.31)
In the other limit of M > 1 we expect the magnetic viscosity to dominate
and the flow to be determined almost entirely by the E x B drift. If we
approximate v(z) for z < a and M > 1, we obtain
V(Z)  ^2 + ( Vl _ £Eo\ e Mz/a (1Q 32)
B \ B Q J
318
Classical Electrodynamics
m:»i
Fig. 10.2 Velocity profiles for
large and small Hartmann
numbers M. For M *■ 0, lami
nar flow occurs. ForMg> l,the
flow is given by the E x B drift
velocity, except in the immediate
neighborhood of the boundaries.
This shows that, while v(z) = V x exactly at the surface, there is a rapid
transition in a distance of order (a/M) to the E x B drift value (cEJBq).
Near z = a, (10.32) is changed by replacing V 1 by V 2 and z by (a — z). The
velocity profile in the two limits (10.31) and (10.32) is shown in Fig. 10.2.
The magnetic field B x (z) is determined by the equation
dB x _ 4tt _ Arra .
oz c c
o B o y )
C I
(10.33)
The boundary conditions on B x at z = and z — a are indeterminate
unless we know the detailed history of how the steady state was created or
can use some symmetry argument. All we know is that the difference in
B x is related to the total current flowing in the y direction per unit length
in the x direction :
B x (a)  B x (0)
c Jo
(z) dz
(10.34)
This indeterminacy stems from the onedimensional nature of the problem.
For simplicity we will calculate the magnetic field only for the case when
the total current in the y direction is zero. * Then we can assume that B x
vanishes at z = and z — a. Using (10.30) for the velocity in (10.33), it
is easy to show that then
BJz) = B
/ 47rora 2
)(*=*)]
cosh cosh
2
(M Mz\
M sinh —
2
(10.35)
* This requirement means that cE /B = ^{V x + V 2 ).
[Sect. 10.4]
Magnetohydrodynamics and Plasma Physics
319
The dimensionless coefficient in square brackets in (10.35) may be identified
as the magnetic Reynolds number (10.15), since (V 2 — V^/2 is a typical
velocity in the problem and a is a typical length. In the two limits M < 1
and M > 1, (10.35) reduces to
BM ~ R M B,
a\ al
l[l(e
.ML \
Mz
+ e
m(2=£)
)]
for M < 1
for M > 1
(10.36)
Figure 10.3 shows the behavior of the lines of force in the two limiting
cases. Only for large R M is there appreciable transport of the lines of force.
And for a given R M , the transport is less the larger the Hartmann number.
For liquid mercury at room temperature the relevant physical constants
are
a = 9.4 x 10 15 sec 1
r\ — 1.5 X 10 2 poise
p = 13.5 gm/cm 3
The diffusion time (10.12) is t = 1.31 x 10" 4 [L (cm)] 2 sec. The Hartmann
number (10.29) is M = 2.64 X lO" 2 ^ (gauss) a (cm). With L ~ a ~ 1 cm,
this gives a magnetic Reynolds number R M ~ IGr^V. Consequently
unless the flow velocity is very large, there is no significant transport of
lines of force for laboratory experiments with mercury. On the other hand,
if the magnetic induction B is of the order of 10 4 gauss, then M ~ 250 and
the velocity flow is almost completely specified by the E x B drift (10.14).
^V 2 >Vi
M«\
M»l
(a)
(b)
Fig. 10.3 (a) Axial component of magnetic induction between the boundary surfaces
for large and small Hartmann numbers. (6) Transport of lines of magnetic induction in
direction of flow.
320 Classical Electrodynamics
In geomagnetic problems with the earth's core and in astrophysical
problems the parameters (e.g., the length scale) are such that R M > 1
occurs often and transport of the lines of force becomes very important.
10.5 Pinch Effect
The confinement of a plasma or conducting fluid by selfmagnetic fields
is of considerable interest in thermonuclear research, as well as in other
applications. To illustrate the principles we consider an infinite cylinder
of conducting fluid with an axial current density J z = J(r) and a resulting
azimuthal magnetic induction B+ = B{r). For simplicity, the current
density, magnetic field, pressure, etc., are assumed to depend only on the
distance r from the cylinder axis, and viscous and gravitational effects are
neglected. We first ask whether a steadystate condition can exist in which
the material is mainly confined within a certain radius r = R by the
action of its own magnetic induction. For a steady state with v = the
equation of motion (10.23) of the fluid reduces to
n dp d(B 2 \ B 2
Ampere's law in integral form relates B{r) to the current enclosed:
4tt f r
B(r) = — rJ(r) dr (10.38)
cr Jo
A number of results can be obtained without specifying the form of J(r),
aside from physical limitations of finiteness, etc. From Ampere's law it is
evident that, if the fluid lies almost entirely inside r = R, then the mag
netic induction outside the fluid is
B(r) = — (10.39)
cr
where
I
R
27rrJ(r) dr
is the total current flowing in the cylinder. Equation (10.37) can be
written as
dp Id, 2r ,<K
Jr = ~^7r (rB) (la40 >
with the solution :
P(r) = Poj~\ r ~ (r 2 5 2 ) dr (10.41)
8tt Jo r 2 dr
[Sect. 10.5] Magnetohydrodynamics and Plasma Physics 321
Here p is the pressure of the fluid at r = 0. If the matter is confined to
r < R, the pressure drops to zero at r = R. Consequently the axial
pressure p is given by
p° = v r L A^ B2)dr (i ° 42)
Stt Jo r* dr
The upper limit of integration can be replaced by infinity, since the inte
grand vanishes for r > R, as can be seen from (10.39). With this expression
(10.42) forp , (10.41) can be written as
p(r) = f \ R \l(r*J?)dr (10.43)
877 Jr r* dr
The average pressure inside the cylinder can be related to the total
current / and radius R without specifying the detailed radial behavior.
Thus
<P>=^ 2 \ B rp(r)dr (10.44)
ttR 2 Jo
Integration by parts and use of (10.40) gives
<P> = — ^— (10.45)
as the relation between average pressure, total current, and radius of the
cylinder of fluid or plasma confined by its own magnetic field. Note that
the average pressure of the matter is equal to the magnetic pressure (B 2 I$tt)
at the surface of the cylinder. In thermonuclear work, hot plasmas with
temperatures of the order of 10 8 °K (kT~ lOkev) and densities of the
order of 10 15 particles/cm 3 are envisioned. These conditions correspond
to a pressure of approximately 10 15 x I0 8 k ^ 1.4 x 10 7 dynes/cm 2 , or 14
atmospheres. A magnetic induction of approximately 19 kilogauss at the
surface, corresponding to a current of 9 x 10 4 /? (cm) amperes, is necessary
for confinement. This shows that extremely high currents are needed to
confine very hot plasmas.
So far the radial behavior of the system has not been discussed. Two
simple examples will serve to illustrate the possibilities. One is that the
current density J(r) is constant for r < R. Then B(r) = (llr/cR 2 ) for
r < R. Equation (10.43) then yields a parabolic dependence for pressure
versus radius :
^^f 1 ?) (ia46)
The axial pressure p is then twice the average pressure (p). The radial
dependences of the various quantities are sketched in Fig. 10.4.
322
Classical Electrodynamics
Fig. 10.4 Variation of azimuthal
magnetic induction and pressure with
radius in a cylindrical plasma column
with a uniform current density /.
The other model has the current density confined to a very thin layer on
the surface, as is appropriate for a highly conducting fluid or plasma. The
magnetic induction is given by (10.39) for r > R, but vanishes inside the
cylinder. Then the pressure p is constant inside the cylinder and equal to
the value (10.45). This is sketched in Fig. 10.5.
10.6 Dynamic Model of the Pinch Effect
The simple considerations of the previous section are valid for a static
or quasistatic situation. In actual practice with plasmas, such circum
stances do not arise. Generally, at some time early in the history of
current flow down the plasma the pressure/? is much too small to resist the
magnetic pressure outside. Consequently the radius of the cylinder of
plasma is forced inwards; the plasma column is pinched. This has the
desirable consequence that the plasma is pulled away from its confining
walls. If the pinched configuration were stable for a sufficiently long time,
it would be possible to heat the plasma to very high temperatures without
burning up the walls of the confining vessel.
Fig. 10.5. Variation of azi
muthal magnetic induction and
pressure in a cylindrical plasma
column with a surfacecurrent
density.
[Sect. 10.6]
Magnetohydrodynamics and Plasma Physics
323
A simple model, first discussed by M. Rosenbluth, exhibits the essential
dynamical features. Suppose that a plasma is created in a hollow con
ducting cylinder of radius Rq and length L. A voltage difference V is
applied between the ends of the cylinder so that a current J flows in the
plasma. This produces an azimuthal magnetic induction B^ which causes
the plasma to pinch inwards. The radius of the plasma column at time
t > is R(t). The conductivity of the plasma is taken to be virtually
infinite. Then the current all flows on the surface, and the magnetic
induction
B+ =  (10.47)
cr
exists only between r = R(t) and r = Rq. Because of the assumption of
infinite conductivity the electric field at the plasma surface, in the moving
frame of reference in which the interface is at rest, vanishes :
E' = E +  xB =
(10.48)
If we now apply Faraday's law of induction to the dotted loop shown in
Fig. 10.6, the inner arm of which is moving inwards with the interface, we
find that the only contribution to the line integral of E comes from the
side of the loop in the conducting wall. Thus
__!i (\drl±(im*9) (10.49)
cdtJmt)* c 2 dt\ R/ v '
This is the standard inductive relation between current, voltage, and
dimensions (inductance). The integral of this equation is
ta (£)W>*
(10.50)
where E f(f) = V/L is the applied electric field. The function fit) is
assumed known and is normalized so that E is the peak value of applied
$ = o *= v
Fig. 10.6 Plasma column inside a hollow,
cylindrical conductor.
■< L *►
324 Classical Electrodynamics
field. In order to proceed further we must relate the current I'm a dynamic
way to the behavior of the plasma radius R.
The desired dynamical connection between I and R is essentially the
momentumbalance equation, or Newton's second law. Some assumption
about the plasma must be made. If the mean free path for collisions is
short compared to the radius, the dynamic behavior is characteristic of
hydrodynamic shock waves. But for a hot, tenuous plasma the mean free
path is comparable to, or larger than, the radius. Then a model with
particles moving freely inside the plasma is more appropriate. If the
velocity A of the plasma surface is large compared to thermal speeds, each
particle approaches the interface with a velocity A in the frame of reference
in which the interface is at rest. As the particle penetrates into the outer
region, it starts feeling the magnetic induction, is turned around, and leaves
the surface with velocity A. Consequently each particle colliding with the
plasma surface receives a momentum transfer 2MA. The number colliding
with unit area of the surface per unit time is NR, where N is the initial
number of particles per unit volume. Therefore the rate of transfer of
momentum per unit area (i.e., pressure) is
p = 2NMR* = 2 P R 2 (10.51)
where p is the initial mass density. At the surface of the plasma there is a
magnetic pressure CB 2 /8tt) due to the discontinuity in magnetic induction
from zero inside to the value B just outside. These pressures must balance.
Consequently, using (10.47), we find that the current is related to the
velocity by :
7 2 = 4"PC 2 R 2 (^f (10.52)
Equation (10.52) depends on a rather simplified model of the mechanical
momentum transfer rate in which each particle collides only once with the
interface. In fact, the velocity of the interface increases with time so that
the surface catches up with particles which were reflected earlier and hits
them again and again. This effect can be approximated by the "snow
plow" model in which the interface is imagined to carry along with it all
the material which it hits as it moves in. Then the magnetic pressure and
rate of change of momentum are related by
d
dtl
M(R)A
d2
■2ttR— (10.53)
07T
where M(R) is the mass carried along by the snowplow:
M(R) = tt P {R*  i?2) ( 10 .54)
[Sect. 10.6]
Magnetohydrodynamics and Plasma Physics
325
This leads to the relation
/2 = npc^R
dtl
(Ro 2 R 2 )f A
(10.55)
between current and radius. In the initial stages when R < R the snow
plow model and freeparticle model give the same relation between current
and radius to within a factor of 2*, and do not differ by an order of
magnitude even at later times.
The equation of motion for R(t) is obtained by substituting P from
either (10.52) or (10.55) into the inductive relation (10.50). Choosing the
freeparticle model as an illustration, we obtain
2R In ( )— =  £&= [fit') df (10.56)
\ R 1 dt y/Anp Jo
where the signs of the square root have been taken to give R < 0. Without
knowledge of f(t) we cannot solve this equation. Nevertheless, some
idea of the solution can be obtained by introducing the dimensionless
variables :
Wl "
\ 47773/ R
_ R_
X ~ R.
Then (10.56) becomes
2x\nx— = [ fir') dr'
dr Jo
For the snowplow model the equivalent equation is
4
drl
<*<
(10.57)
(10.58)
(10.59)
M
(10.60)
z(ln zf
Without solving these equations it is evident that x changes significantly
in times such that t ~ 1. This means that the scaling law for the radial
velocity of the pinch is
\Airp J
This result emerges whatever dynamic model is used, including a hydro
dynamic one. Typical experimental conditions for a fast pinch in small
scale hydrogen or deuterium plasmas involve applied electric fields of the
* The factor of 2 comes from the fact that in the one case the particles are elastically
reflected and suffer a velocity change of 2R, while in the other the particles collide
inelastically with the interface and receive a velocity change of R.
326
Classical Electrodynamics
Fig. 10.7 Radius of plasma
column as a function of time
after initiation of current flow.
The characteristic velocity of
pinching is given by (10.60).
order of 10 3 volts/cm and initial densities of the order of 10 8 gm/cm 3
(~ 3 X 10 15 deuterons/cm 3 ). Then v is of the order of 10 7 cm/sec. The
current flowing is, according to (10.52) or (10.55),
v n \ dr)
(10.61)
where F is a dimensionless function of the order of unity. For a tube
radius of 10 cm and the conditions described, the current /is measured in
units of 10 5 or 10 6 amperes.
The discussion of the pinching action presented so far is obviously valid
only for short times after the initiation of current flow. The simplified
models indicate that in a time of the order of R /v the radius of the plasma
column goes to zero. It is clear, however, that before that will occur (even
approximately) the behavior will be modified. In the hydrodynamic limit,
the radial shock waves caused by the pinch will be reflected off the axis
and move outwards, striking the interface and retarding its inward motion
or even reversing it. This phenomenon is known as bouncing. It is
evidently present also in the freeparticle model. Consequently the general
behavior of radius R as a function of time is expected to be as shown in
Fig. 10.7. Although no proper analysis has been made of the subsequent
bounces, it is conjectured that there is an approach to a steady state at
some radius less than R .
10.7 Instabilities in a PinchedPIasma Column
In the laboratory longlived pinched plasmas are extremely difficult to
produce. The dynamic behavior of the previous section is found to be
followed at least qualitatively for times up to around the first bounce.
But then the plasma column is observed to break up rapidly. The reason
for the disintegration of the column is the growth of instabilities. The
[Sect. 10.7]
Magnetohydrodynamics and Plasma Physics
327
Fig. 10.8 (a) Kink instability.
(6) Sausage or neck instability.
(b)
column is unstable against various departures from cylindrical geometry.
Small distortions are amplified rapidly and destroy the column in a very
short time. The detailed analysis of instabilities is sufficiently complex
that we will attempt only qualitative arguments. Two of the simpler
unstable distortions will be described.
The first is the kink instability, shown in Fig. 10.8a. The lines of azimu
thal magnetic induction near the column are bunched together above, and
separated below, the column by the distortion downwards. Thus the
magnetic pressure changes are in such a direction as to increase the
distortion. The distortion is unstable.
The second type of distortion is called a sausage or neck instability,
shown in Fig. 10.86. In the neighborhood of the constriction the azi
muthal induction increases, causing a greater inwards pressure at the neck
than elsewhere. This serves to enhance the existing distortion.
Both types of instability are hindered by axial magnetic fields within
the plasma column. For the sausage distortion the lines of axial induction
are compressed by the constriction, causing an increased pressure inside
to oppose the increased pressure of the azimuthal field, as indicated
schematically in Fig. 10.9. It is easy to see that the fractional changes in
Fig. 10.9 Hindering neck instability with
outward pressure of trapped axial magnetic
fields.
328
Classical Electrodynamics
Fig. 10.10 Hindering kink instability with tension of trapped axial fields.
the two magnetic pressures, assuming a sharp boundary to the plasma,
are A ~ A
Ap^ = 2x Ap z _ 4x
P4> ~ R ' p z R
where x is the small inwards displacement. Consequently, if
(10.62)
B? > *V
(10.63)
the column is stable against sausage distortions.
For kinks the axial magnetic field lines are stretched, rather than com
pressed laterally together. But the result is the same ; namely the increased
tension in the field lines inside opposes the external forces and tends to
stabilize the column. It is evident from Fig. 10.10 that a short wavelength
kink of a given lateral displacement will cause the lines of force to stretch
relatively more than a longwavelength kink. Consequently, for a given
ratio of internal axial field to external azimuthal field, there will be a
tendency to stabilize shortwavelength kinks, but not very longwavelength
ones. If the fields are approximately equal, analysis shows that if the wave
length of the kink X < 14 R the disturbance is stabilized.
For longerwavelength kinks stabilization can be achieved by the action
of the outer conductor, provided the plasma radius is not too small
compared to the radius of the conductor. The azimuthal field lines are
trapped between the conductor and the plasma boundary, as shown in
Fig. 10.11. If the plasma column moves too close to the walls, the lines
of force are crowded together between it and the walls, causing an in
creased magnetic pressure and restoring force.
Fig. 10.11 Stabilization of
longwavelength kinks with
outer conductor.
[Sect. 10.8] Magnetohydrodynamics and Plasma Physics 329
It is clear qualitatively that it must be possible, by a combination of
trapped axial field and conducting walls, to create a stable configuration,
at least in the approximation of a highly conducting plasma with a sharp
boundary. Detailed analysis* confirms this qualitative conclusion and
sets limits on the quantities involved. It is important to have as little axial
field outside the plasma as possible and to keep the plasma radius of the
order of onehalf or onethird of the cylinder radius. If the axial field
outside the plasma is too large, the combined B z and B+ cause helical
instabilities that are troublesome in toroidal geometries. If, however, the
axial field outside the plasma is made very large, the pitch of the helix
becomes so great that there is much less than one turn of the helix in a
plasma column of finite length. Then it turns out that there is the possi
bility of stability again. Stabilization by means of a strong axial field
produced by currents external to the plasma is the basis of some fusion
devices, e.g., the Stellarator.
The idealized situation of a sharp plasma boundary is difficult to create
experimentally, and even when created is destroyed by diffusion of the
plasma through the lines of force in times of the order of AttgR 2 \c 2 (see
Section 10.3). For a hydrogen plasma of 1 ev energy per particle this time
is of the order of 10 4 sec for R ~ 10 cm, while for a 10 kev plasma it is
of the order of 10 2 sec. Clearly the thermonuclear experimenter must try
to create initially as hot a plasma as possible in order to make the initial
diffusion time long enough to allow further heating.
10.8 Magnetohydrodynamic Waves
In ordinary hydrodynamics the only type of smallamplitude wave
motion possible is that of longitudinal, compressional (sound) waves.
These propagate with a velocity s related to the derivative of pressure with
respect to density at constant entropy:
Wo
(10.64)
dp/
If the adiabatic law p = Kp y is assumed, s 2 = yp lp , where y is the ratio
of specific heats. In magnetohydrodynamics another type of wave motion
is possible. It is associated with the transverse motion of lines of magnetic
force. The tension in the lines of force tends to restore them to straight
line form, thereby causing a transverse oscillation. By analogy with
* V. D. Shafranov, Atomnaya Energ. I, 5, 38 (1956); R. J. Tayler, Proc. Phys. Soc.
{London), B70, 1049 (1957); M. Rosenbluth, Los Alamos Report L/l2030 (1956). See
also Proceedings of the Second International Conference on Peaceful Uses of Atomic
Energy, Vol. 31 (1958), papers by Braginsky and Shafranov (p. 43) and Tayler (p. 160).
330
Classical Electrodynamics
ordinary sound waves whose velocity squared is of the order of the hydro
static pressure divided by the density, we expect that these magnetohydro
dynamic waves, called Alfven waves, will have a velocity
v A ~(p\ (10.65)
where B 2 I&it is the magnetic pressure.
To examine the wave motion of a conducting fluid in the presence of a
magnetic field, we consider a compressible, nonviscous, perfectly con
ducting fluid in a magnetic field in the absence of gravitational forces. The
appropriate equations governing its behavior are :
dt
(py) = o
p^+/>(v.V)v= v P
at
— = V x (v x B)
dt
— B x (V x B)
4tt
(10.66)
These must be supplemented by an equation of state relating the pressure
to the density. We assume that the equilibrium velocity is zero, but that
there exists a spatially uniform, static, magnetic induction B throughout
the uniform fluid of constant density p . Then we imagine smallamplitude^
departures from the equilibrium values :
B = B o + B 1 (x,0"
P = Po + fti(x, ► 00.67)
v = Vi(x, J
If equations (10.66) are linearized in the small quantities, then they
become :
dpi
Bt
+ / , V.v 1 =
Po ^ + s*V Pl + ^x(VxB 1 ) =
dt Att
dBi
dt
 V x ( Vl x B ) =
(10.68)
where s 2 is the square of the sound velocity (10.64). These equations can
be combined to yield an equation for v x alone :
?^ _ S 2 V (V . Vl ) + v^ x V x [V x ( Vl x yj] = (10.69)
dt 2
[Sect. 10.8] Magnetohydrodynamics and Plasma Physics 331
where we have introduced a vectorial Alfven velocity:
y A = ^= (10.70)
The wave equation (10.69) for v x is somewhat involved, but it allows
simple solutions for waves propagating parallel or perpendicular to the
magnetic field direction.* With v 1 (x, t) a plane wave with wave vector k
and frequency co :
Vl (x, /) = Vie**"** (10.71)
equation (10.69) becomes:
a> 2 v 1 + (.s 2 + VX k 'Vi)k
+ v A • k[(v^ • k)v x  {y A • v x )k  (k • Vl )vJ = (10.72)
If k is perpendicular to v A the last term vanishes. Then the solution for v x
is a longitudinal magnetosonic wave with a phase velocity :
"long = V* 2 + v A * (10.73)
Note that this wave propagates with a velocity which depends on the sum
of hydrostatic and magnetic pressures, apart from factors of the order of
unity. If k is parallel to \ A , (10.72) reduces to
(5)
(*V  <>vi + [f 2  1 J k\y A • Vl )v, = (10.74)
There are two types of wave motion possible in this case. There is an
ordinary longitudinal wave (v x parallel to k and v^) with phase velocity
equal to the sound velocity s. But there is also a transverse wave (v x • v A =
0) with a phase velocity equal to the Alfven velocity v A . This Alfven wave
is a purely magnetohydrodynamic phenomenon which depends only on
the magnetic field (tension) and the density (inertia).
For mercury at room temperature the Alfven velocity is [B (gauss)/ 13.1])
cm/sec, compared with sound speed of 1.45 x 10 5 cm/sec. At all labora
tory field strengths the Alfven velocity is much less than the speed of
sound. In astrophysical problems, on the other hand, the Alfven velocity
can become very large because of the much smaller densities. In the sun's
photosphere, for example, the density is of the order of 10~ 7 gm/cm 3
(~6 x 10 16 hydrogen atoms/cm 3 ) so that v A ^ 10 3 B cm/sec. Solar
magnetic fields appear to be of the order of 1 or 2 gauss at the surface, with
much larger values around sunspots. For comparison, the velocity of
sound is of the order of 10 6 cm/sec in both the photosphere and the
chromosphere.
* The determination of the characteristics of the waves for arbitrary direction of
propagation is left to Problem 10.3.
332
Classical Electrodynamics
,* n mm
m m M'*A
/^ M M M M An
A h M A II >lM
MM A A
An A m
(a) (6)
Fig. 10.12 Magnetohydrodynamic waves.
The magnetic fields of these different waves can be found from the third
equation in (10.68):
r k
B x =
«A
for klB
for the longitudinal k  B
5 V 1
ft)
for the transverse k  B
(10.75)
The magnetosonic wave moving perpendicular to B causes compressions
and rarefactions in the lines of force without changing their direction, as
indicated in Fig. 10. 12a. The Alfven wave parallel to B causes the lines of
force to oscillate back and forth laterally (Fig. 10.126). In either case the
lines of force are "frozen in" and move with the fluid.
If the conductivity of the fluid is not infinite or viscous effects are present,
we anticipate dissipative losses and a consequent damping of oscillations.
The second and third equations in (10.68) are modified by additional terms :
Po ?ll = _*» V Pl  5° x (V x B x ) + r, V\
Ot 47T
^i = V x( Vl xB ) + /V 2 B 1
dt 4tto
(10.76)
where r\ is the viscosity* and a is the conductivity. Since both additions
cause dispersion in the phase velocity, their effects are most easily seen
when a plane wave solution is being sought. For plane waves it is evident
* Use of the simple viscous force (10.3) is not really allowed for a compressible fluid.
But it can be expected to give the correct qualitative behavior.
[Sect. 10.8] Magnetohydrodynamics and Plasma Physics 333
that these equations are equivalent to
dy 1 1
dt
\ Po (o/
dB t 1
s*V Pl ^ x(V xBO
V x ( Vl x B )
(10.77)
Consequently equation (10.72) relating k and a> is modified by (a) multi
/ c z k 2 \
plying s 2 and to 2 by the factor I 1 + / 1 , and (b) multiplying a> 2 by
/ k 2\ \ AttOW}
the factor 1 + i — .
\ p Q M/
For the important case of the Alfven wave parallel to the field, the
relation between a> and k becomes
fcV = ^ 2 (l + * —) (l + * —) (10.78)
\ AttO(jo' \ PqO)'
If the resistive and viscous correction terms are small, the wave number is
approximately
k ~«L +t MjL + 3\ (10.79)
v A 2vJ\4na Po /
This shows that the attenuation increases rapidly with frequency (or wave
number), but decreases with increasing magnetic field strength. In terms
of the diffusion time t of Section 10.3, the imaginary part of the wave
number shows that, apart from viscosity effects, the wave travels for a time
t before falling to 1/e of its original intensity, where the length parameter
in t (10.12) is the wavelength of oscillation. For the opposite extreme in
which the resistive and/or viscous terms dominate, the wave number is
given by the vanishing of the two factors on the righthand side of (10.78).
Thus k has equal real and imaginary parts and the wave is damped out
rapidly, independent of the magnitude of the magnetic field.
The considerations of magnetohydrodynamic waves given above are
valid only at comparatively low frequencies, since the displacement
current was ignored in Ampere's law. It is evident that, if the frequency is
high enough, the behavior of the fields must go over into the "ionospheric"
behavior described in Section 7.9, where chargeseparation effects play an
important role. But even when chargeseparation effects are neglected in
the magnetohydrodynamic description, the displacement current modifies
the propagation of the Alfven and magnetosonic waves. The form of
334 Classical Electrodynamics
Ampere's law, including the displacement current, is :
VxB = J,(vxB) (10.80)
c c 2 dt
where we have used the infinite conductivity approximation (10.9) in
eliminating the electric field E. Thus the current to be inserted into the
force equation for fluid motion is now
J = —
47T
VxB +  2 (vxB)
c 2 dt
(10.81)
In the linearized set of equations (10.68) the second one is then generalized
to read:
Po
£ + M§"*)]— ' Vft 5" <y,, * ) (1082)
'Ldt
This means that the wave equation for v x is altered to the form:
\(i + v 4)H(^^)
dfL x \ c 2 / c
s 2 V(V. Vl )
+ T i xVxVx(T 1 xyJ = (10.83)
Inspection shows that for y 1 parallel to v A (i.e., B ) there is no change from
before. But for transverse ^ (either magnetosonic with k perpendicular
to B , or Alfven waves with k parallel to B ) the square of the frequency is
multiplied by a factor [1 + (v A 2 lc 2 )]. Thus the phase velocity of Alfven
waves becomes
u A = , CVa (10.84)
Vc 2 + vj
In the usual limit where v A < c, the velocity is approximately equal to v A ;
the displacement current is unimportant. But, if v A > c, then the phase
velocity is equal to the velocity of light. From the point of view of electro
magnetic waves, the transverse Alfven wave can be thought of as a wave
in a medium with an index of refraction given by
Thus
u A =  (10.85)
n
n 2 = 1 + — = 1 + ^4 < ia86 >
Caution must be urged in using this index of refraction for the propagation
of electromagnetic waves in a plasma. It is valid only at frequencies where
chargeseparation effects are unimportant.
[Sect. 10.9] Magnetohydrodynamics and Plasma Physics 335
10.9 HighFrequency Plasma Oscillations
The magnetohydrodynamic approximation considered in the previous
sections is based on the concept of a singlecomponent, electrically neutral
fluid with a scalar conductivity a to describe its interaction with the
electromagnetic field. As discussed in the introduction to this chapter a
conducting fluid or plasma is, however, a multicomponent fluid with
electrons and one or more types of ions present. At low frequencies or
long wavelengths the description in terms of a single fluid is valid because
the collision frequency v is large enough (and the mean free path short
enough) that the electrons and ions always maintain local electrical
neutrality, while on the average drifting in opposite directions according
to Ohm's law under the action of electric fields. At higher frequencies the
singlefluid model breaks down. The electrons and ions tend to move
independently, and charge separations occur. These charge separations
produce strong restoring forces. Consequently oscillations of an electro
static nature are set up. If a magnetic field is present, other effects occur.
The electrons and ions tend to move in circular or helical orbits in the
magnetic field with orbital frequencies given by
co B = — (10.87)
mc
When the fields are strong enough or the densities low enough that the
orbital frequencies are comparable to the collision frequency, the concept
of a scalar conductivity breaks down and the current flow exhibits a
marked directional dependence relative to the magnetic field (see Problem
10.5). At still higher frequencies the greater inertia of the ions implies that
they will be unable to follow the rapid fluctuation of the fields. Only the
electrons partake in the motion. The ions merely provide a uniform back
ground of positive charge to give electrical neutrality on the average. The
idea of a uniform background of charge, and indeed the concept of an
electron fluid, is valid only when we are considering a scale of length which
is at least large compared to interparticle spacings (/ > n ~ lyi ). In fact, there
is another limit, the Debye screening length, which for plasmas at reasonable
temperatures is greater than « ~^, and which forms the actual dividing
line between smallscale individualparticle motion and collective fluid
motion (see the following section).
To avoid undue complications we consider only the highfrequency
behavior of a plasma, ignoring the dynamical effects of the ions. We also
ignore the effects of collisions. The electrons of charge e and mass m are
336
Classical Electrodynamics
described by a density n(x, t) and an average velocity v(x, t). The equi
libriumcharge density of ions and electrons is ^en^. The dynamical
equations for the electron fluid are
dt
(m) =
By
Bt
+ (vV)v
m \
E +  x B
c
^v P
mn
(10.88)
where the effects of the thermal kinetic energy of the electrons are described
by the electron pressure p (here assumed a scalar). The charge and
current densities are :
p =e(nn )\
(10.89)
J = ens
Thus Maxwell's
equation
s can be written
V
E = 4nein — n )
V
•B =
V
c dt
V
_ 1 dE Anen
x B =
c dt c
(10.90)
We now assume that the static situation is the electron fluid at rest with
n — n and no fields present, and consider small departures from that
state due to some initial disturbance. The linearized equations of motion
are
dt
mn \on/o
d\ e
dt m
V • E  Anen =
47ren
U =
(10.91)
VxBi^
c dt
plus the two homogeneous Maxwell's equations. Here «(x, t) and v(x, t)
represent departures from equilibrium. If an external magnetic field B
is present a [(v/c) x B ] term must be kept in the force equation (see
Problem 10.7), but the fluctuation field B is of first order in small quantities
so that (v x B) is second order. The continuity equation is actually not
[Sect. 10.9] Magnetohydrodynamics and Plasma Physics 337
an independent equation, but may be derived by combining the last two
equations in (10.91).
Since the force equation in (10.91) is independent of magnetic field, we
suspect that there exist solutions of a purely electrostatic nature, with
B = 0. The continuity and force equations can be combined to yield a
wave equation for the density fluctuations :
?» + l*z£»*\ n _ 1 (?P\ V 2 n = (10 .92)
dt 2 \ m / m \9n/o
On the other hand, the time derivative of Ampere's law and the force
equation can be combined to give an equation for the fields :
^E /W«p\ E _ 1 (dp\ V(y . E) = cV x » (10 .93)
dt 2 \ m 1 m \dnJo ot
The structures of the lefthand sides of these two equations are essentially
identical. Consequently no inconsistency arises if we put dB/dt = 0.
Having excluded static fields already, we conclude that B = is a possi
bility. If dB/dt = 0, then Faraday's law implies V x E = 0. Hence E is
a longitudinal field derivable from a scalar potential. It is immediately
evident that each component of E satisfies the same equation (10.92) as the
density fluctuations. If the pressure term in (10.92) is neglected, we find
that the density, velocity, and electric field all oscillate with the plasma
frequency (o p :
2 _ 47rn o g (10.94)
«V =
m
If the pressure term is included, we obtain a dispersion relation for the
frequency :
co 2 = co/ +  &) k 2 (10.95)
m \dn/o
The determination of the coefficient of k 2 takes some care. The adiabatic
law p = p (nln o y can be assumed, but the customary acoustical value
y = f for a gas of particles with 3 external, but no internal, degrees of
freedom is not valid. The reason is that the frequency of the present
density oscillations is much higher than the collision frequency, contrary
to the acoustical limit. Consequently the onedimensional nature of the
density oscillations is maintained. A value of y appropriate to 1 trans
lational degree of freedom must be used. Since y = (m + 2)//w, where m
is the number of degrees of freedom, we have in this case y = 3. Then
i (?p) _
m \dn'o
3 P»_ (10.96)
mn
338
Classical Electrodynamics
If we use/? = n KT and define the rms velocity component in one direction
(parallel to the electric field),
m(u 2 ) = KT
then the dispersion equation can be written
a, 2 = co p 2 + 3(u 2 )k 2
(10.97)
(10.98)
This relation is an approximate one, valid for long wavelengths, and is
actually just the first two terms in an expansion involving higher and
higher moments of the velocity distribution of the electrons (see Problem
10.6). In form (10.98) the dispersion equation has a validity beyond the
ideal gas law which was used in the derivation. For example, it applies to
plasma oscillations in a degenerate Fermi gas of electrons in which all cells
in velocity space are filled inside a sphere of radius equal to the Fermi
velocity V F . Then the average value of the square of a component of
velocity is
< M 2 > = lV r * (10.99)
Quantum effects appear explicitly in the dispersion equation only in higher
order terms in the expansion in powers of k 2 .
The oscillations described above are longitudinal electrostatic oscilla
tions in which the oscillating magnetic field vanishes identically. This
means that they cannot give rise to radiation in an unbounded plasma.
There are, however, modes of oscillation in a plasma which are transverse
electromagnetic waves. To see the various possibilities of plasma oscil
lations we assume that all variables vary as exp (ik • x — icot) and look for
a defining relationship between co and k, as we did for the magnetohydro
dynamic waves in Section 10.8. With this assumption the linearized
equations (10.91) and the two homogeneous Maxwell's equations can be
written :
kv
n = n
ieE , 3<w 2 > n .
mco co n„
'0
k • E = — iArren
kB =
k x B =
— E — i ^v
co
k x E = B
c
(10.100)
[Sect. 10.10] Magnetohydrodynamics and Plasma Physics 339
Maxwell's equations can be solved for v in terms of k and E :
v = (^) —,  [(co 2  c 2 /c 2 )E + c 2 (k • E)k] (10.101)
\mtol co p
Then the force equation and the divergence of E can be used to eliminate v
in order to obtain an equation for E alone :
(co 2  co/  c 2 fc 2 )E + (c 2  3<M 2 »(k • E)k = (10.102)
If we write E in terms of components parallel and perpendicular to k:
E = Ej f Ej_
where E„ = H^r) 1
then (10.102) can be written as two equations:
(co 2  co/  3<« 2 )A: 2 )E, =
(10.103)
(10.104)
(co 2  ft) p 2  c 2 & 2 )Ej. = '
The first of these results shows that the longitudinal waves satisfy the
dispersion relation (10.98) already discussed, while the second shows that
there are two transverse waves (two states of polarization) which have the
dispersion relation :
co 2 = co/ + c 2 A: 2 (10.105)
Equation (10.105) is just the dispersion equation for the transverse
electromagnetic waves described in Section 7.9 from another point of view.
In the absence of external fields the electrostatic oscillations and the trans
verse electromagnetic oscillations are not coupled together. But in the
presence of an external magnetic induction, for example, the force equa
tion has an added term involving the magnetic field and the oscillations
are coupled (see Problem 10.7).
10.10 ShortWavelength Limit for Plasma Oscillations and the
Debye Screening Distance
In the discussion of plasma oscillations so far no mention has been made
of the range of wave numbers over which the description in terms of
collective oscillations applies. Certainly n£ is one upper bound on the
wavenumber scale. A clue to a more relevant upper bound can be
obtained by examining the dispersion relation (10.98) for the longitudinal
oscillations. For long wavelengths the frequency of oscillation is very
340
Classical Electrodynamics
closely co = co p . It is only for wave numbers comparable to the Debye
wave number k
d>
k D * =
<u 2 >
(10.106)
that appreciable departures of the frequency from co p occur.
For wave numbers k < k D , the phase and group velocities of the
longitudinal plasma oscillations are :
k
3<« 2 >
(10.107)
From the definition of k D we see that for such wave numbers the phase
velocity is much larger than, and the group velocity much smaller than,
the rms thermal velocity (u 2 ) 1A . As the wave number increases towards
k D , the phase velocity decreases from large values down towards (u 2 ) 1/4 .
Consequently for wave numbers of the order of k D the wave travels with
a small enough velocity that there are appreciable numbers of electrons
traveling somewhat faster than, or slower than, or at about the same speed
as, the wave. The phase velocity lies in the tail of the thermal distribution.
The circumstance that the wave's velocity is comparable with the electronic
thermal velocities is the source of an energytransfer mechanism which
causes the destruction of the oscillation. The mechanism is the trapping
of particles by the moving wave with a resultant transfer of energy out of
the wave motion into the particles. The consequent damping of the wave
is called Landau damping.
A detailed calculation of Landau damping is out of place here. But we
can describe qualitatively the physical mechanism. Fig. 10.13 shows a
distribution of electron velocities with a certain rms spread and a
Maxwellian tail out to higher velocities. For small k the phase velocity
no(v)
Fig. 10.13 Thermal velocity
distribution of electrons.
[Sect. 10.10] Magnetohydrodynamics and Plasma Physics 341
lies far out on the tail and negligible damping occurs. But as k > k D the
phase velocity lies within the tail, as shown in Fig. 10.13, with a significant
number of electrons having thermal speeds comparable to v p . There is
then a velocity band Ay around v = v p where electrons are moving
sufficiently slowly relative to the wave that they can be trapped in the
potential troughs and carried along at velocity v p by the wave. If there are
more particles in At; moving initially slower than v p than there are
particles moving faster (as shown in the figure), the trapping process will
cause a net increase in the energy of the particles at the expense of the
wave. This is the mechanism of Landau damping. Detailed calculations
show that the damping can be expressed in terms of an imaginary part of
the frequency given by
Im co ~ co p J" (^>Je«» 2 <^ (10.108)
provided k<k D . To obtain (10.108) a Maxwellian distribution of
velocities was assumed. For k >k D the damping constant is larger than
given by (10.108) and rapidly becomes much larger than the real part of
the frequency, as given by (10.98).
The Landau formula (10.108) shows that for k<k D the longitudinal
plasma oscillations are virtually undamped. But the damping becomes
important as soon as k~k D (even for k = 0.5^, Im co ~ — 0Jco p ).
For wave numbers larger than the Debye wave number the damping is so
great that it is meaningless to speak of organized oscillations.
Another, rather different consideration leads to the same limiting Debye
wave number as the boundary of collective oscillatory effects. We know
that an electronic plasma is a collection of electrons with a uniform back
ground of positive charge. On a very small scale of length we must
describe the behavior in terms of a succession of very many twobody
Coulomb collisions. But on a larger scale the electrons tend to cooperate.
If a local surplus of positive charge appears anywhere, the electrons rush
over to neutralize it. This collective response to charge fluctuations is
what gives rise to largescale plasma oscillations. But in addition to, or,
better, because of, the collective oscillations the cooperative response of
the electrons also tends to reduce the longrange nature of the Coulomb
interaction between particles. An individual electron is, after all, a local
fluctuation in the charge density. The surrounding electrons are repelled
in such a way that they tend to screen out the Coulomb field of the chosen
electron, turning it into a shortrange interaction. That something like
this must occur is obvious when one realizes that the only source of
electrostatic interaction is the Coulomb force between the particles. If
some of it is effectively taken away to cause longwavelength collective
342 Classical Electrodynamics
plasma oscillations, the residue must be a sum of shortrange interactions
between particles.
A nonrigorous derivation of the screening effect described above was
first given by Debye and Hiickel in their theory of electrolytes. The basic
argument is as follows. Suppose that we have a plasma with a distribution
of electrons in thermal equilibrium in an electrostatic potential O. Then
they are distributed according to the Boltzmann factor e ~ H/KT where His
the electronic Hamiltonian. The spatial density of electrons is therefore
n(x) = n eW KT > (10.109)
Now we imagine a test charge Ze placed at the origin in this distribution
of electrons with its uniform background of positive ions (charge density
— en ). The resulting potential <I> will be determined by Poisson's equation
V 2 = AirZe <5(x)  4>nen Q [e^ IKT )  1] (10.110)
If (e<S>/KT) is assumed small, the equation can be linearized:
V a O  k 2 D ® ~ AirZe d(x) (10. 1 1 1)
where
, 2 _ 47rn e 2 /inii<y>
k d — KT (10.112)
is an alternative way of writing (10.106). Equation (10.111) has the
spherically symmetric solution :
k D r
<&(r) = Ze (10.113)
r
showing that the electrons move in such a way as to screen out the Coulomb
field of a test charge in a distance of the order of k D ~ x . The balance between
thermal kinetic energy and electrostatic energy determines the magnitude
of the screening radius. Numerically
\n /
cm (10.114)
where T is in degrees Kelvin, and n is the number of electrons per cubic
centimeter. For a typical hot plasma with T = 10 6 °^and n = 10 15 cm 3 ,
we find kjf 1 ^ 2.2 x 10~ 4 cm.
For the degenerate electron gas at low temperatures the Debye wave
number k D is replaced by a Fermi wave number k F :
k F ~^f (10.115)
where V F is the velocity at the surface of the Fermi sphere. This magni
tude of screening radius can be deduced from a FermiThomas generaliza
tion of the DebyeHuckel approach. It fits in naturally with the dispersion
relation (10.98) and the mean square velocity (10.99).
[Probs. 10] Magnetohydrodynamics and Plasma Physics 343
The DebyeHuckel screening distance provides a natural dividing line
between the smallscale collisions of pairs of particles and the largescale
collective effects such as plasma oscillations. It is a happy and not
fortuitous happening that plasma oscillations of shorter wavelengths can
independently be shown not to exist because of severe damping.
REFERENCES AND SUGGESTED READING
The subject of magnetohydrodynamics and plasma physics has a rapidly growing
literature. Many of the available books are collections of papers presented at confer
ences and symposia. Although these are useful to someone who has some knowledge of
the field, they are not suited for beginning study. Two works on magnetohydrodynamics
which are coherent presentations of the subject are
Alfv6n,
Cowling.
A short discussion of magnetohydrodynamics appears in
Landau and Lifshitz, Electrodynamics of Continuous Media, Chapter VIII.
Corresponding books devoted mainly to the physics of plasmas are
Chandrasekhar,
Linhart,
Simon,
Spitzer.
The subject of controlled thermonuclear reactions, with much material on the funda
mental physics of plasmas, is treated thoroughly by
Glasstone and Lovberg,
Rose and Clark.
PROBLEMS
10.1 An infinitely long, solid, right circular, metallic cylinder has radius (R/2)
and conductivity a. It is tightly surrounded by, but insulated from, a hollow
cylinder of the same material of inner radius (R/2) and outer radius R.
Equal and opposite total currents, distributed uniformly over the cross
sectional areas, flow in the inner cylinder and in the hollow outer one.
At t = the applied voltages are shortcircuited.
(a) Find the distribution of magnetic induction inside the cylinders
before t = 0.
(b) Find the distribution as a function of time after t = 0, neglecting
the displacement current.
(c) What is the behavior of the magnetic induction as a function of time
for long times? Define what you mean by long times.
10.2 A comparatively stable selfpinched column of plasma can be produced by
trapping an axial magnetic induction inside the plasma before the pinch
begins. Suppose that the plasma column initially fills a conducting tube of
radius R and that a uniform axial magnetic induction B z0 is present in the
344 Classical Electrodynamics
tube. Then a voltage is applied along the tube so that axial currents flow
and an azimuthal magnetic induction is built up.
(a) Show that, if quasiequilibrium conditions apply, the pressurebalance
relation can be written :
pir) + £l + £*!
1 ["Eldr
+ — t± dr =
(b) If the plasma has a sharp boundary and such a large conductivity
that currents flow only in a thin layer on the surface, show that for a
quasistatic situation the radius R(t) of the plasma column is given by the
equation
where
W£)f.I>*
BmRft
'0 —
cE
and E f(t) is the applied electric field.
(c) If the initial axial field is 100 gauss, and the applied electric field has
an initial value of 1 volt/cm and falls almost linearly to zero in 1 millisecond,
determine the final radius if the initial radius is 50 cm. These conditions are
of the same order of magnitude as those appropriate for the British toroidal
apparatus (Zeta), but external inductive effects limit the pinching effect to
less than the value found here. See E. P. Butt et al., Proceedings of the
Second International Conference on Peaceful Uses of Atomic Energy, Vol. 32,
p. 42 (1958).
10.3 Magnetohydrodynamic waves can occur in a compressible, nonviscous,
perfectly conducting fluid in a uniform static magnetic induction B . If the
propagation direction is not parallel or perpendicular to B , the waves are
not separated into purely longitudinal (magnetosonic) or transverse (Alfven)
waves. Let the angle between the propagation direction k and the field B
bed.
(a) Show that there are three different waves with phase velocities given by
«i 2 = (Vx cos 0) 2
"1,3 = K* 2 + *>a 2 ) ± U(s 2 + va 2 ) 2  4sW cos 2 6}V2
where s is the sound velocity in the fluid, and va. = (B 2 /47rp o yA is the
Alfven velocity.
(b) Find the velocity eigenvectors for the three different waves, and
prove that the first (Alfven) wave is always transverse, while the other two
are neither longitudinal nor transverse.
(c) Evaluate the phase velocities and eigenvectors of the mixed waves in
the approximation that vj, :> s. Show that for one wave the only appreciable
component of velocity is parallel to the magnetic field, while for the other
the only component is perpendicular to the field and in the plane containing
k and B .
10.4 An incompressible, nonviscous, perfectly conducting fluid with constant
density p is acted upon by a gravitational potential y> and a uniform, static,
magnetic induction B .
[Probs. 10] Magnetohydrodynamics and Plasma Physics 345
(a) Show that magnetohydrodynamic waves of arbitrary amplitude and
form BjCx, 0, v(x, t) can exist, described by the equations
(B V)B 1 = ±V47r Po — x
B x = ±V4^v
, (B + B x ) 2 t +
P + PaW H ^ — — = constant
(Z>) Suppose that at r = a certain disturbance B^x, 0) exists in the
fluid such that it satisfies the above equations with the upper sign. What
is the behavior of the disturbance at later times ?
10.5 The force equation for an electronic plasma, including a phenomenological
collision term, but neglecting the hydrostatic pressure (zero temperature
approximation) is
^ + (v . V )v =U+ xb) v
dt m\ c J
where v is the collision frequency.
(a) Show that in the presence of static, uniform, external, electric, and
magnetic fields, the linearized steadystate expression for Ohm's law
becomes
i
where the conductivity tensor is
1 ^
,2\ V
4ttvI 1 +  B '
\ v
i +^4
and co v (to B ) is the electronic plasma (precession) frequency. The direction
of B is chosen as the z axis.
(b) Suppose that at t = an external electric field E is suddenly applied
in the x direction, there being a magnetic induction B in the z direction.
The current is zero at t = 0. Find expressions for the components of the
current at all times, including the transient behavior.
10.6 The effects of finite temperature on a plasma can be described approxi
mately by means of the correlationless Boltzmann (Vlasov) equation. Let
f(x, v, t) be the distribution function for electrons of charge e and mass m
in a onecomponent plasma. The Vlasov equation is
f=f + v.v,/ + a.V„/ =
where V^ and V v are gradients with respect to coordinate and velocity, and
a is the acceleration of a particle. For electrostatic oscillations of the
346 Classical Electrodynamics
plasma a = eE/m, where E is the macroscopic electric field satisfying
V • E = 4ne I /(x, v, /) (Pv  n
If / (v) is the normalized equilibrium distribution of electrons
/ (v) d 3 v = n
"o /o(
(a) show that the dispersion relation for smallamplitude longitudinal
plasma oscillations is
co/ Jkvo)
(b) assuming that the phase velocity of the wave is large compared to
thermal velocities, show that the dispersion relation gives
0)2 i ^^< k * v > , , <(kv) 2 > ,
5 ^ 1 + l + 5 5 — + • • •
co/ co or
where < > means averaged over the equilibrium distribution / (v). Relate
this result to that obtained in the text with the electronic fluid model.
(c) What is the meaning of the singularity in the dispersion relation when
kv = co?
10.7 Consider the problem of waves in an electronic plasma when an external
magnetic field B is present. Use the fluid model, neglecting the pressure
term as well as collisions.
(a) Write down the linearized equations of motion and Maxwell's
equations, assuming all variables vary as exp (ik '• x — ia>t).
(b) Show that the dispersion relation for the frequencies of the different
modes in terms of the wave number can be written
co 2 (co 2  VX^ 2  V  * 2c2 ) 2
= oo B 2 (a>*  Ar 2 c 2 )[o> 2 (co 2  co/  A: 2 c 2 ) + co/c^k • b) 2 ]
where b is a unit vector in the direction of B ; co v and a> B are the plasma
and precession frequencies, respectively.
(c) Assuming co B <; a> p , solve approximately for the various roots for
the cases (i) k parallel to b, (ii) k perpendicular to b. Sketch your results
for co 2 versus k 2 in the two cases.
11
Special Theory of Relativity
The special theory of relativity has been treated extensively in many
books. Its history is interwoven with the history of electromagnetism. In
fact, one can say that the development of Maxwell's equations with the
unification of electricity and magnetism and optics forced special relativity
on us. Lorentz laid the groundwork in his studies of electrodynamics,
while Einstein contributed crucial concepts and placed the theory on a
consistent and general basis. Even though special relativity had its origin
in electromagnetism and optics, it is now believed to apply to all
types of interactions except, of course, largescale gravitational phenomena.
In modern physics the theory serves as a touchstone for possible forms for
the interactions between elementary particles. Only theories consistent
with special relativity need to be considered. This often severely limits the
possibilities. Since the experimental basis and the development of the
theory are described in detail in many places, we will content ourselves
with a summary of the key points.
11.1 Historical Background and Key Experiments
In the forty years before 1900 electromagnetism and optics were cor
related and explained in triumphal fashion by the wave theory based on
Maxwell's equations. Since previous experience with wave motion had
always involved a medium for the propagation of waves, it was natural for
physicists to assume that light needed a medium through which to propa
gate. In view of the known facts about light it was necessary to assume
that this medium, called the ether, permeated all space, was of negligible
density, and had negligible interaction with matter. It existed solely as a
vehicle for the propagation of electromagnetic waves. The hypothesis of
347
348 Classical Electrodynamics
an ether set electromagnetic phenomena apart from the rest of physics,
For a long time it had been known that the laws of mechanics were the
same in different coordinate systems moving uniformly relative to one
another — the laws of mechanics are invariant under Galilean transfor
mations. The existence of an ether implied that the laws of electro
magnetism were not invariant under Galilean coordinate transformations.
There was a preferred coordinate system in which the ether was at rest.
There the velocity of light in vacuum was equal to c. In other coordinate
frames the velocity of light was presumably not c.
To avoid setting electromagnetism apart from the rest of physics by a
failure of Galilean relativity there are several avenues open. Some of these
are:
1. Assume that the velocity of light is equal to c with respect to a
coordinate system in which the source is at rest.
2. Assume that the preferred reference frame for light is the coordinate
system in which the medium through which the light is propagating is at
rest.
3. Assume that, although the ether has a very small interaction with
matter, the interaction is enough that it can be carried along with astro
nomical bodies such as the earth.
A large number of experiments led to the abandonment of all these
hypotheses and the birth of the special theory of relativity. Three basic
experiments are :
(1) Observation of the aberration of star positions during the year,
(2) Fizeau's experiment on the velocity of light in moving fluids (1859),
(3) Michel sonMorley experiment to detect motion through the ether
(1887).
The aberration of star light (the small shift in apparent position of
distant stars during the year) is an ancient phenomenon which finds a
simple explanation in the motion of our earth in its orbit around the sun
at a velocity of the order of 3 x 10 6 cm/sec. Suppose that the star light
is incident normal to the earth's surface while the velocity of the earth in
orbit is parallel to the surface. Figure 11.1 shows that a telescope must be
inclined at an angle a, where
v
a
~  ~ 1(T 4 radian (11.1)
c
in order that the light pass down it to the observer as the telescope moves
along. Six months later the velocity vector v will be in the opposite
direction. The star will then appear at an angle a on the other side of the
vertical. The apparent positions of stars trace out elliptical paths on the
celestial sphere during the year with angular spreads of the order of (11.1).
[Sect. 11.1]
Special Theory of Relativity
349
Fig. 11.1 Aberration of star positions.
This simple explanation of aberration contradicts the hypothesis that the
velocity of light is determined by the transmitting medium (our atmosphere
in this case) or that the ether is dragged along by the earth. In neither case
would aberration occur.
Fizeau's experiment involved measuring, by means of an interferometer,
the velocity of light in liquids flowing in a pipe, both in the direction of and
opposed to the propagation of the light. If the index of refraction of the
liquid is n, then depending on which of the various hypotheses one
chooses, he expects the velocity to be
c
u = ,
n
n
(11.2)
where v is the velocity of flow. The actual result found by Fizeau was,
within experimental error,
B £ ± ,(i_l)
n \ n I
(11.3)
We note that this result can be made consistent with the ether being dragged
along by the earth only by assuming that smaller bodies are partially
successful in carrying the ether with them. Even then the assumption is
rather artificial in that their effectiveness at carrying the ether involves
their indices of refraction. *
The MichelsonMorley experiment was designed to detect a motion of
the earth relative to a preferred reference frame (the ether at rest) in which
the velocity of light is c. The basic apparatus is shown schematically in
Fig. 1 1.2. A laboratory light source S is focused on a thinly silvered glass
plate P which divides the light into two beams at right angles to each other,
one of which goes to mirror M x and is reflected back through the plate
* Actually formula (11.3) is a theoretical one proposed in 1818 by Fresnel on the basis
of a model in which the density of the elastic ether in matter is proportional to n 2 .
350
Classical Electrodynamics
B 2 By
Fig. 11.2 MichelsonMorley experiment.
to B x and the other of which goes to mirror M 2 , back to the plate, and is
reflected to B 2 . Conditions are such that the two beams travel almost the
same path length. Small differences in path length or in the times taken to
traverse the paths are detected by observing shifts in interference fringes
produced by the two beams. The whole apparatus was attached to a stone
slab floating in mercury so that it could be rotated. Suppose that velocity
v of the earth through the ether is parallel to the light path from P to M 2 .
Then the velocity of light relative to the apparatus on the path from P to
M 2 and return is c ± v. If the path distance from P to M 2 is d 2 , the time
taken by the light to go from P to M 2 and return is
4— + T)*
\c — V C + VI c
1
(3
(11.4)
For the path from P to M x and return it is convenient to view things from
the preferred coordinate frame. Then it is evident that the path length
traversed is greater than d lf the distance from P to M x , because the mirror
is moving with velocity v through the ether.
Figure 11.3 shows the geometrical relations. Evidently sin a = v/c, so
that the effective path length is
2d x sec a = 2d x
1
i4
c
and the time taken is
2d x 1
(11.5)
[Sect. 11.1] Special Theory of Relativity
The difference between the two transit times is
At = t 2 — t x = 
d 2
c
k
v 2
1
f7 2
1
J
—
c 2 '
V
c 2 l
351
(11.6)
If we assume that v < c, we can expand the denominators, obtaining
At
(11.7)
If the apparatus is now rotated through 90°, the transit times become
H —
c
f , 2d i
h —
c
1
J*$
(11.8)
1
i4
and the difference, to lowest order, is
At'  ?[(<*,  d x ) + ^(  ^)] (11.7')
Since At and At' are not the same, we expect a shift in the interference
fringes upon rotation of the apparatus, the shift being proportional to
t
„2
At' — At =
1 17"
(di + dj
c c l
(11.9)
Since the orbital velocity of the earth is about 3 X 10 6 cm/sec, we expect
v 2 /c z ~ 10~ 8 , at least at some time of the year. With (d x + d^ ~ 3 x 10 2
cm, the time difference (11.9) is 10 16 sec. This means that the relevant
length (to be compared to a wavelength of light) is c \At' — Ar . — ' 3 x
10 6 cm = 300 A. Since visible light has wavelengths of the order of
CZZ.3
Fig. 11.3
352 Classical Electrodynamics
3000 A, the expected effect is a fringe shift of about onetenth of a fringe.
The accuracy of the MichelsonMorley experiment was such that a
relative velocity of 10 6 cm/sec would have been seen (i.e., onethird of the
above estimate). No fringe shift was found. Since the original work of
Michelson the experiment has been repeated many times with modifi
cations such as very unequal path lengths. No evidence for relative motion
through the ether has been found. A summary of all the available evidence
has been given by Shankland et ah, Revs. Modern Phys., 27, 167 (1955).
The negative result of the MichelsonMorley experiment can be ex
plained on the etherdrag hypothesis. But that hypothesis is inconsistent
with the aberration of star light. Only the socalled emission theories,
where the velocity of light is constant relative to the source, are consistent
with all three of the experiments cited. And we will see in the next section
that other experiments exclude such theories. On the positive side the
MichelsonMorley experiment can be thought of as restoring electro
magnetism to the rest of physics in the matter of relativity. No observable
effects were found which depended on the motion of the apparatus
relative to some conjectured absolute reference frame. It should be
mentioned, however, that FitzGerald and Lorentz (1892) explained the
null result while still retaining the ether concept by postulating that all
material objects are contracted in their direction of motion as they move
through the ether. The rule of contraction is
UP)= L 0a/ 1_ ~2 (1L10)
It is clear from (11.4) or (11.7) that this hypothesis leads to a zero result
for (At' — At) in place of (11.9). The FitzGeraldLorentz contraction
hypothesis was perhaps the last gasp of the ether advocates, and it contains
the germ of the special theory of relativity. The contraction phenomenon
is present in special relativity, but in a more general way applying to all
systems in relative motion with one another. Going along with it is the
phenomenon of time dilatation (not postulated by FitzGerald or Lorentz),
an experimentally wellfounded effect. These are discussed in Section 1 1.3.
11.2 The Postulates of Special Relativity and the Lorentz
Transformation
In 1904 Lorentz showed that Maxwell's equations in vacuum were
invariant under a transformation of coordinates given by (11.19) below,
and now called a Lorentz transformation, provided the field strengths were
[Sect. 11.2] Special Theory of Relativity 353
suitably transformed. By supposing that all matter was essentially electro
magnetic in origin and so transformed in the same way as Maxwell's
equations, Lorentz was able to deduce the contraction law (11.10). Then
Poincare showed that the transformation of charge and current densities
could be made in such a way that all the equations of electrodynamics are
invariant in form under Lorentz transformations. In 1905, almost at the
same time as Poincare and without knowledge of Lorentz's paper, Einstein
formulated special relativity in a general and complete way, obtaining the
results of Lorentz and Poincare, but showing that the ideas were of much
wider applicability. Instead of basing his discussion on electrodynamics,
Einstein showed that just two postulates were necessary, one of them
involving only a very general property of light.
The two postulates of Einstein were :
1 . POSTULAtE OF RELATIVITY
The laws of nature and the results of all experiments performed
in a given frame of reference are independent of the translational
motion of the system as a whole. Thus there exists a triply infinite
set of equivalent reference frames moving with constant velocities
in rectilinear paths relative to one another in which all physical
phenomena occur in an identical manner.
For brevity these equivalent coordinate systems are called Galilean
reference frames. The postulate of relativity is consistent with all our
experience with mechanics where only relative motion between bodies is
relevant. It is also consistent with the MichelsonMorley experiment and
makes meaningless the question of detecting motion relative to the ether.
2. POSTULATE OF THE CONSTANCY OF THE VELOCITY OF LIGHT
The velocity of light is independent of the motion of the source.
This hypothesis, untested when Einstein proposed it, is necessary and
decisive in obtaining the Lorentz transformation and all its consequences
(see below). Because our classical concept of time as a variable independent
of the spatial coordinates is destroyed by this postulate, its acceptance was
resisted vehemently for a number of years. Many ingenious attempts were
made to invent theories which would explain all the observed facts without
this assumption. One notable one was Ritz's version of electrodynamics,
which kept the two homogeneous Maxwell's equations intact but modified
the equations involving the sources in such a way that the velocity of light
was equal to c only when measured relative to the source. Experiments
have proved all such theories wrong and have established the constancy
of the velocity of light independent of the motion of the source. One such
experiment is the MichelsonMorley interferometer experiment performed
354
Classical Electrodynamics
o\ ) >
Fig. 11.4
with starlight, rather than a terrestrial light source. No effect was observed
which could be attributed to a change in the velocity of light due to the
relative motion of the star and the earth. Another experiment on the light
from rotating binary stars showed that the velocity of light depends
negligibly (if c' = c + kv, then k < 0.002) on the motion of the stars
toward or away from us.
The constancy of the velocity of light, independent of the motion of the
source, allows us to deduce the connection between spacetime coordinates
in different Galilean reference frames. To see how this is possible we
consider two coordinate systems K and K'. System K' has its axes parallel
to those of K, but it is moving with a velocity v in the positive z direction
relative to the system K, as shown in Fig. 11.4. Points in space and time
in the two systems are specified by (x, y, z, t) and (x', y', z', t), respectively.
For convenience we suppose that a common origin of time t — t ' = is
chosen at the instant when the two sets of coordinate axes exactly overlap.
Now we imagine an observer in each reference frame equipped with the
necessary apparatus (e.g., a network of correlated clocks and photocells
at known distances from the origin) to detect the arrival time of a light
signal from the origin at various points in space. If there is a light source
at rest in the system K (and so moving with velocity v in the negative z
direction in system K') which is flashed on and off rapidly at t = t' = 0,
then Einstein's second postulate implies that each observer will see his
photocell network respond to a spherical shell of radiation moving out
ward from his origin of coordinates with velocity c. Consequently the
arrival time / of the pulse at a detector located at (x, y, z) in system K will
satisfy the equation :
x 2 + y 2 + z 2  c 2 t 2 =
Similarly, in system K' the wave front is described by
x' 2 + y' 2 + z' 2  cH' 2 =
(11.11)
(11.12)
[Sect.11.2]
Special Theory of Relativity
355
Relations (11.11) and (11.12) seem to violate the first postulate of relativity.
If two observers in different coordinate systems both see spherical pulses
centered on a fixed origin in each system, the spheres must be different!
This apparent contradiction is resolved when we allow the possibility that
events which are simultaneous in one coordinate system are not necessarily
simultaneous in another coordinate system moving relative to the first. We
can now anticipate that time is no longer an absolute quantity independent
of spatial variables and of relative motion.
To obtain a connection between the coordinates (x\ y', z', t') of system
K' and (x, y, z, i) of system Kit is only necessary to assume that the trans
formation is linear. This seems very plausible and is equivalent to the
assumption that spacetime is homogeneous and isotropic. If the trans
formation is linear, the only possible connection between the quadratic
forms (11.11) and (11.12) is
z' 2 + y' 2 + z' 2  c 2 t' 2 = X\x 2 + y* + z 2 c 2 t 2 ) (11.13)
where X = X{v) with X(0) = 1. The presence of X allows for the possibility
of an overall scale change in going from K to K'. But shells of radiation
are spheres in both systems. From the hypothesis that K' is moving
parallel to the z axis of K, it is evident that the transformation of x', y'
must be
x' = Xx, y' = Xy (11.14)
independent of the time, because motion parallel to the z axis in K' must
remain so in K. Then the most general linear connection between z', t'
and z, t is
z' = X{a x z + a 2 t), t' = X(b x t + b#) (11.15)
A factor X has been extracted for convenience. The coefficients a x , a 2 , b lt b 2
are functions of v with the following limiting values as v — ► :
«i
V
im .
a%
 — •
;+0
h
l
A.
0,
(11.16)
The origin of K' moves with a velocity v in the system K. Consequently
its position is specified by z = vt. This means that a 2 = —va x in (11.15).
If equations (11.15) are now substituted into (11.13), three algebraic
relations between a x , b x , and b 2 are obtained. These are easily solved to
356 Classical Electrodynamics
give the following values, with signs chosen to agree with (11.16):
, 1
a x = b x =
l V 
c 2
(11.17)
C 2
There remains the problem of the determination of X(v). If a third
reference frame K" is considered to be moving with a velocity — v parallel
to the z axis relative to K\ the coordinates (x", y", z", t") can be obtained
in terms of (x r , y', z', t') from the above results merely by the change
v>—v. But the system K" is just the original system K, so that x" = x,
y" = y, z" = z, t" = t. This leads to the requirement that
X(v) X{v) = 1
(11.18)
But X{v) must be independent of the sign of v, since it represents a scale
change in the transverse direction. Consequently X(v) = 1. Then we can
write down the Lorentz transformation, connecting coordinates in K' to
those in K:
t' =
z — vt
„2
C 2
x = x, y = y
(11.19)
Transformation (11.19) represents a special case in which the relative
motion of K and K' is parallel to the z axis. It is a straightforward matter
to write down the result for an arbitrary velocity v of translation of K'
relative to K, as shown in Fig. 11.5. Equation (11.19) clearly applies to
parallel and perpendicular components of the coordinates relative to v:
x„ =
1
(x„  vf), x/ = x ±
c 2
j>4 cj
(11.20)
[Sect. 11.3]
Special Theory of Relativity
357
Fig. 11.5
With the definition that x„ = [(v • x)v]/y 2 and x ± = x — x„, equations
(11.20) can be combined to yield the general Lorentz transformation:*
1 Axv 1
x' = x +
 1
vt
c 2
M<  M
l ~*
(11.21)
It should be noted that (11.21) represents a single Lorentz transformation
to a reference frame K' moving with velocity v relative to the system K.
Successive Lorentz transformations do not in general commute. It is easy
to show that they commute only if the successive velocities are parallel.
Consequently three successive transformations corresponding to the com
ponents of the velocity v in three mutually perpendicular directions yield
different results, depending on the order in which the transformations are
applied, and none agrees with (11.21). (See Problem 11.2.)
11.3 FitzGeraldLorentz Contraction and Time Dilatation
As has already been mentioned, FitzGerald and Lorentz proposed the
contraction rule (11.10) for the dimensions of an object parallel to its
* The word "general" is not really applicable to transformation (1 1 .21). The connota
tion here is that the direction of the velocity v is arbitrary. But a more general trans
formation would allow the axes in K' to be rotated relative to those in, K. Even this
Lorentz transformation is not the most general, since it is still homogeneous in the co
ordinates. The general inhomogeneous Lorentz transformation allows translation of the
origin in spacetime as well. See M0ller, Section 18.
358 Classical Electrodynamics
Fig. 11.6 Moving rod. FitzGeraldLorentz
contraction.
motion at velocity v through the ether; Lorentz was able to give the rule
an electrodynamic basis from the properties of Maxwell's equations under
Lorentz transformations. We now show that the contraction of lengths
in the direction of motion is a more general phenomenon which applies to
all relative motion. Consider a rod of length L at rest parallel to the z'
axis in the system K' of the previous section, as indicated schematically in
Fig. 1 1.6. By definition L = z 2 ' — z/, where z/ and z 2 ' are the coordinates
of the end points of the rod in K'. In the system A:the length L of the rod is,
again by definition, L = z 2 — z x , where z 2 and z 1 are the instantaneous
coordinates of the end points of the rod, observed at the same time t.
From (11.19) the length in K' is
L = V  z x > = Z )~ % \ = == (11.22)
which is just the FitzGeraldLorentz result (11.10). Note that in the
system K the length is defined at equal times t. The fact that this is not at
equal times in the system K' is not relevant for the definition of length in
the system K. This again illustrates that simultaneity is only a relative
concept.
Another consequence of the special theory of relativity is time dilatation.
A clock moving relative to an observer is found to run more slowly than
one at rest relative to him. The most fundamental "clocks" which we
have available are the unstable elementary particles. Each type of particle
decays at rest with a welldefined lifetime (mean life) which is unaltered by
external fields, apart from nuclear or atomic force fields which cause
transformations that are well understood.* These particles can serve
therefore as "clocks" which can be examined at rest and in motion.
Suppose that we consider a meson of lifetime t at rest in the system K'
* For example, negative mu mesons can become bound in hydrogenlike orbits
around nuclei with binding energies that are not negligible compared to the energy
liberated in their decay. Since the rate of decay depends sensitively on the energy release
(closely as the fifth power of AE), tightly bound negative mu mesons exhibit a consider
ably slower rate of decay than unbound ones.
[Sect. 11.3] Special Theory of Relativity 359
which is moving with uniform velocity v relative to the system K. We
assume that the meson is created at the origin of K' at time t' = t = 0. As
seen from the system K the position of the meson is given by z = vt. If it
lives a time t in K', then at its instant of decay, we find
v
t z
c 2 L v 2
i v 
''=o = l^= = 'Jl 9 (U 23 )
c
The time / is the meson's lifetime t as observed in the system K. Con
sequently
T ° (11.24)
c 2
When viewed from K the moving meson lives longer than a meson at rest
in K. The "clock" in motion is observed to run more slowly than an
identical one at rest.
Time dilatation has been observed in cosmic rays with highenergy mu
mesons. These mesons are produced as secondary particles at a height of
the order of 10 or 20 km, and a large fraction of them reach the earth's
surface. Since the mean lifetime of a mu meson is t = 2.2 x 10~ 6 sec,
it could travel no more than ct = 0.66 km on the average before decaying
if no time dilatation occurred. Clearly dilatation factors of the order of
10 or more are involved, consistent with the high energies (velocities
approaching the velocity of light) of these particles.
A laboratory experiment exhibiting time dilatation with pi mesons is
not difficult to perform. Charged pi mesons have a mean lifetime t =
2.56 x 10~ 8 sec. An experiment studying the numbers of charged pi
mesons decaying in flight per unit length as a function of distance from the
point of production was done at Columbia University. * The mesons had
a velocity v ~ 0.75c. The numbers of mesons decaying per unit distance
should follow an exponential law N(x) = N e~ x/X , where A is the mean
free path in the laboratory and x is the distance from the source (corrected
for finite solid angles, etc.). Figure 1 1.7 shows schematically the results of
the experiment. The mean free path is X = 8.5 ± 0.6 meters. Since
X = vt, the laboratory lifetime is r = 3.8 ± 0.3 x 10 _8 sec. Consequently
r 2.56
* Durbin, Loar, and Havens, Phys. Rev., 88, 179 (1952). This experiment was per
formed to measure the lifetime of the pi meson. The validity of time dilatation was
assumed. But with independent knowledge of the lifetime, the argument can be inverted
as we do here.
360
Classical Electrodynamics
X obs = 8.5+0.6
meters
Fig. 11.7 Number of pi mesons decaying
per unit distance as a function of distance
from the point of production (schematic).
This value is to be compared with 1.51 calculated from (11.24) with
v = 0.75c. The laboratory experiment on time dilatation is not so
dramatic as the cosmic rays but has the great virtue of being performed
under controlled conditions in a comparatively small space.
11.4 Addition of Velocities: Aberration and the Fizeau Experiment;
Doppler Shift
The Lorentz transformation (11.19) can be used to obtain the addition
law for velocities. Suppose that there is a velocity vector u' in the system
K' which makes polar angles 6', <f>' with the z' axis as shown in Fig. 11.8.
The system K' is moving relative to the system K with a velocity v in the z
direction. We want to know the components of velocity u as seen in the
system K. From (11.19), or rather the inverse transformation, the dif
ferential expressions for dx, dy, dz, dt can be obtained:
dx = dx', dy = dy'
1
dz =
(dz' + v dt'), dt =
c 2
. 1 Lt' + dz) 1(11
25)
This means that the components of velocity are
u„ =
1   "a
c 2
u z ' + v
1  1 <
c l
1 +
VIA,
(11.26)
[Sect. 11.4]
Special Theory of Relativity
361
The magnitude of u and the angles 6, <f> of u in the system K are easily
found. Since u x 'lu y ' = uju y , the azimuthal angles are equal, <f> = <f>'.
Similarly
tan0 =
1^
u sin
c 2 u' cos 6' + v
and
+ v 2 + 2u'v cos 6' —
u v sin
1 +
u ' v a>\
cose)
(11.27)
The inverse results for (u', 0') in terms of (u, 6) can be obtained by inter
changing «<> u', 6<^ 6', and also changing the sign of v.
Study of (11.26) or (11.27) shows that for velocities u' and v both small
compared to c the addition law is just that of Galilean relativity, u = u' f
v. But for either velocity approaching that of light, modifications appear.
It is impossible to obtain a velocity greater than that of light by adding
two velocities, even if each is very close to c. For the simple case of parallel
velocities the addition law is
+ v
u =
(11.28)
1 +
If u' = c, then (11.28) shows that u = c also. This is merely an explicit
statement of Einstein's second postulate.
The laws of addition of velocities are in accord with both the aberration
of starlight and the Fizeau experiment. For the aberration, the velocity u'
Fig. 11.8 Addition of velocities.
362 Classical Electrodynamics
is that of light in the system K', namely, u' = c, and v is the orbital velocity
of the earth. Then the angle 6 is related to d' by
«» 9 v i 4 H!lL i (11  29)
c cos 6' + 
c
For starlight incident normally on the earth 6' = tt/2. Then
*™ d = C Ji v 2 0130)
The angle 6 is the complement of the angle a in (11.1). Thus
tan a = —ML= (11.31)
$
completely consistent with observation. (Since vjc ~ 10 4 , the departure
of the radical from unity is far beyond the realm of observation.)
In the simplest version of the Fizeau experiment the liquid flows with
velocity v parallel or antiparallel to the path of light. If the liquid has an
index of refraction n we may assume that light propagates with a velocity
u' = c/n relative to the liquid. From (1 1 .28) the velocity of light observed
in the laboratory is
C ±v
u =   ± vl 1  U (11.32)
nc
The latter expression is the expansion to lowest order of the exact result.
This is in agreement with the Fresnel formula (1 1.3). Actually there is an
added term in (1 1.32) if the index of refraction is a function of wavelength.
It comes about because of the Doppler shift in wavelength in the moving
liquid. The increase AA in wavelength in the moving medium is
AA=±An (11.33)
c
correct to lowest order in vjc for the parallel and antiparallel velocities,
respectively. Consequently the appropriate velocity of light in the liquid
is
c c c dn A .
~ — Ax
n(A + AA) n(X) n 2 dk
[Sect. 11.4] Special Theory of Relativity 363
Then the corrected expression to replace (11.32) is
,— e ±„(l !*£) dl34)
The correction due to dispersive effects has been observed.
The relativistic Doppler shift formulas can be obtained from the fact
that the phase of a light wave is an invariant quantity. Actually, the phase of
any plane wave is invariant under a Lorentz transformation, the reason
being that the phase can be associated with mere counting which is
independent of coordinate frame. Consider a plane wave of frequency co
and wave vector k in the reference frame K. An observer at the point P
with coordinate x is equipped to record the number of wave crests which
reach him in a certain time. If the wave crest passing the origin at t = is
the first one which he records (when it reaches him), then at time t he will
have counted
— (k • x — cot)
2tt
wave crests. Now imagine another reference frame K' which moves
relative to the frame K with a velocity v parallel to the z axis, and has its
origin coincident with that of K at t = 0. An observer in K' at the point
P' with coordinate x' is equipped similarly to the one in K. He begins
counting when the wave crest passing the origin reaches him, and con
tinues counting until time t '. If the point P' is such that at the end of the
counting period it coincides with the point P, then both observers must have
counted the same number of wave crests. But the observer in K' has
counted
wave crests, where k' and co' are the wave vector and frequency of the plane
wave in K '. Thus the phase of the wave is invariant. Consequently we
have
k'x'  co't' = kx cot (11.35)
Using the transformation formulas (11.19), we find
v ' — v U .' _ u
■tt'^ "fl
2 («  vk z )
(11.36)
364 Classical Electrodynamics
For light waves, k = cojc and k' = co'/c. Hence these results can be
expressed in the form :
CO
HH
c 2
tan 6' =
1 v 2 ■
1  — sin
c 2
a v
COS
(11.37)
c
where 6 and 6' are the angles of k and k' relative to the direction of v. This
last equation is just the inverse of (11.29).
It is sometimes useful to have the frequency co' expressed in terms of the
angle 0' of the wave in the frame K'. From the inverse of the first equation
in (11.37), it is evident that the desired expression is
i ^
1 co
C (11.38)
(l+COS0'j
The first equation in (11.37) is the customary Doppler shift, modified
by the radical in the denominator. The presence of the square root shows
that there is a transverse Doppler shift, even when 6 = tt\2. This relati
vistic transverse Doppler shift has been observed spectroscopically with
atoms in motion (IvesStilwell experiment, 1938). It also has been observed
using a precise resonanceabsorption technique, with a nuclear gammaray
source on the axis of a rapidly rotating cylinder and the absorber attached
to the circumference of the cylinder.
11.5 Thomas Precession
In 1926, Uhlenbeck and Goudsmit introduced the idea of electron spin
and showed that, if the electron had a g factor of 2, the anomalous Zeeman
effect could be explained, as well as the existence of multiplet splittings.
There was a difficulty, however, in that the observed finestructure intervals
were only onehalf the theoretically expected values. If a g factor of unity
were chosen, the finestructure intervals were given correctly, but the
Zeeman effect was then the normal one. The complete explanation of spin,
including correctly the g factor and the proper finestructure interaction,
came only with the relativistic electron theory of Dirac. But within the
[Sect. 11.5] Special Theory of Relativity 365
framework of an empirical spin angular momentum and a g factor of 2,
Thomas showed that the origin of the discrepancy was a relativistic
kinematic effect which, when included properly, gave both the anomalous
Zeeman effect and the correct finestructure splittings. The Thomas
precession, as it is called, also gives a qualitative explanation for a spin
orbit interaction in atomic nuclei and shows why the doublets are
"inverted" in nuclei.
The UhlenbeckGoudsmit hypothesis was that an electron possessed a
spin angular momentum S (which could take on quantized values of ±h/2
along any axis) and a magnetic moment p related to S by
\l = — S (11.39)
mc
The customary relation between magnetic moment and angular momentum
is given by (5.64). Equation (1 1.39) shows that the electron has a g factor
of 2. Suppose that an electron moves with a velocity v in external fields E
and B. Then the equation of motion for its angular momentum in its rest
frame is
— = {x x B' (11.40)
dt
where B' is the magnetic induction in that frame. We will show in Section
11.10 that in a coordinate system moving with the electron the magnetic
induction is
B'~ <B xEJ (11.41)
where we have neglected terms of the order of (v 2 /c 2 ). Then (11.40)
becomes
— =ix BxE (11.42)
dt \ c I
Equation (11.42) is equivalent to an energy of interaction of the electron
spin:
r= ji. IB x e)
U' = p. IB xEl (11.43)
In an atom the electric force eE can be approximated as the negative
gradient of a spherically symmetric average potential energy V(r). For one
electron atoms this is, of course, exact. Thus
r dV
eE=— (11.44)
r dr
366
Classical Electrodynamics
Fig. 11.9
Then the spininteraction energy can be written
U' =  — S • B + k (S • L)  —
(11.45)
mc m"C r dr
where L = m(r x v) is the electron's orbital angular momentum. This
interaction energy gives the anomalous Zeeman effect correctly, but has a
spinorbit interaction which is twice too large.
The error in (11.45) can be traced to the incorrectness of (11.40) as an
equation of motion for the electron spin. The lefthand side of (11.40)
gives the rate of change of spin in the rest frame of the electron. This is
equal to the applied torque ({jl x B') only if the electron's rest frame is not
a rotating coordinate system. If, as Thomas first pointed out, that co
ordinate system rotates, then the time rate of change of any vector G in that
system is*
dG = ldG\
dt \ dt / nonrot
where <o T is the angular velocity of rotation found by Thomas,
applied to the electron spin, (11.46) gives an equation of motion:
dS =r (eB'
dt
The corresponding energy of interaction is
U=U' Su> T (11.48)
where U' is the electromagnetic spin interaction (11.45).
The origin of the Thomas precessional frequency <a T is the acceleration
experienced by the electron as it moves in its atomic orbit. Figure 11.9
shows the electron at position 1 at a time t with instantaneous velocity
ttirp X G
I + Wyl
\mc 1
(11.46)
When
(11.47)
See, for example, Goldstein, p. 133.
[Sect. 11.5]
Special Theory of Relativity
367
vector v, and at position 2 an infinitesimal time later (/ + bt) with
velocity v + by. The increment in velocity is related to the electron's
acceleration a by <5v = a dt. At time t the electron's rest frame K' and the
laboratory frame K are related by a Lorentz transformation with velocity
v. At time t + bt the electron's rest frame has now changed to K",
related to K by a Lorentz transformation with velocity v + by. The
question now arises, "How are the coordinate frames K" and K' related ?
That is, how do the axes in the electron's rest frame behave in time ?" As
viewed from the laboratory, in a time dt the electron undergoes an infini
tesimal change in velocity <5v. Consequently we might anticipate that K"
and K' would be connected by a simple infinitesimal Lorentz transfor
mation. If so, (11.45) would be correct as it stands. To see that the con
nection is more than a mere Lorentz transformation we note that the
transformation from K' to K" is equivalent to two successive Lorentz
transformations, one with velocity — v, and the other with velocity v + by.
K"
K' ^\\+K— v+ by
(11.49)
Now it is generally true that two successive Lorentz transformations are
equivalent to a single Lorentz transformation plus a rotation. Using the
general formula (11.21) twice, it is a straightforward matter to show that
the time variables in K" and K' are related by
t" = f  A
l
i L ,
c
by +
 1
<5v
(11.50)
correct to first order in 6\. This shows that the direct transformation from
K' to K" involves an infinitesimal Lorentz transformation with a velocity
Av =
1
c 2
<5v +
The corresponding transformation of the coordinates is
x » _ x  + /_i _ A x < x (UL*)
— Avf
(11.51)
(11.52)
Comparison with x" = x' + x' x A£l for a rotation of axes by an in
finitesimal angle ASl shows that the coordinate axes in K" are rotated
relative to those in K' by an angle
Aii = /J_=  l\Ti£ (11.53)
368 Classical Electrodynamics
This shows that the coordinate axes in the electron's rest frame precess
with an angular velocity
where the result on the right is an approximation valid if v < c. We
emphasize the purely kinematic origin of the Thomas precession by noting
that nothing has been said about the cause of the acceleration. If a
component of acceleration exists perpendicular to v, then there is a Thomas
precession, independent of other effects such as precession of the magnetic
moment in a magnetic field.
For electrons in atoms the acceleration is caused by the screened
Coulomb field (1 1.44). Thus the Thomas angular velocity is
_ 1 r x vldV 1 T ldV
to T ~ = L (11.55)
2c 2 m r dr 2m 2 c 2 r dr
It is evident from (11.48) and (11.45) that the extra contribution to the
energy from the Thomas precession just reduces the spinorbit coupling
by a factor of \ (sometimes called the Thomas factor), yielding
C /«_Ls.B+A 1 S.L±^ (11.56)
mc 2m c r dr
as the correct spinorbit interaction energy for an atomic electron.
In atomic nuclei the nucleons experience strong accelerations due to the
specifically nuclear forces. The electromagnetic forces are comparatively
weak. In an approximate way one can treat the nucleons as moving
separately in a shortrange, spherically symmetric, attractive, potential
well, V N (r). Then each nucleon will experience in addition a spinorbit
interaction given by (11.48) with the negligible electromagnetic contri
bution U' omitted :
[/jv^SWj, (11.57)
where the acceleration in to T is determined by V N {r). The form of w T is
the same as (11.55) with V replaced by V N . Thus the nuclear spinorbit
interaction is approximately
U N ~ ^ 9 S.L^v (n.58)
2M 2 c 2 rdr
In comparing (11.58) with atomic formula (11.56) we note that both V
and V N are attractive (although V N is much larger), so that the signs of
the spinorbit energies are opposite. This means that in nuclei the single
particle levels form "inverted" doublets. With a reasonable form for V N ,
[Sect. 11.6] Special Theory of Relativity 369
(11.58) is in qualitative agreement with the observed spinorbit splittings
in nuclei.
11.6 Proper Time and the Light Cone
In the previous sections we have explored some of the physical con
sequences of the special theory of relativity and Lorentz transformations.
In the next two sections we want now to discuss some of the more formal
aspects and to introduce some notation and concepts which are very useful
in a systematic discussion of physical theories within the framework of
special relativity.
In Galilean relativity space and time coordinates are unconnected.
Consequently under Galilean transformations the infinitesimal elements
of distance and time are separately invariant. Thus
ds 2 = dx 2 + dy 2 + dz 2 = ds' 2
dt 2 = dt' 2
(11.59)
For Lorentz transformations, on the other hand, the time and space
coordinates are interrelated. From (11.21) it is easy to show that the
invariant "length" element is
ds 2 = dx 2 + dy 2 + dz 2  c 2 dt 2 (1 1.60)
This leads immediately to the concept of a Lorentz invariant proper time.
Consider a system, which for definiteness we will think of as a particle,
moving with an instantaneous velocity v{f) relative to some coordinate
system K. In the coordinate system K' where the particle is instantaneously
at rest the spacetime increments are dx' = dy' = dz' = 0, dt' = dr. Then
the invariant length (11.60) is
c 2 dr 2 = dx 2 + dy 2 + dz 2  c 2 dt 2 (1 1.61)
In terms of the particle velocity \(t) this can be written
 d 'J l ' I
(11.62)
Equation (11.62) shows the timedilatation effect already discussed. But
much more important, by the manner of its derivation (11.62) shows that
the time t, called the proper time of the particle, is a Lorentz invariant
quantity. This is of considerable importance later on when we wish to
discuss various quantities and their time derivatives. If a quantity behaves
in a certain way under Lorentz transformations, then its proper time
370
Classical Electrodynamics
Fig. 11.10 World line of a system and the
light cone. The unshaded interior of the cone
represents the past and the future, while the
shaded region outside the cone is called
"elsewhere." A point inside (outside) the
light cone is said to have a timelike (space
like) separation from the origin.
derivative will behave in the same way because of the invariance of dr.
But its ordinary time derivative will not have the same transformation
properties. From (11.62) we see that a certain proper time interval
(t 2 — r x ) will be seen in the system K as a time interval
C T 2 dr
= , (11.63)
u =
V\r)
where t x and t 2 are the corresponding times in K.
Another fruitful concept in special relativity is the idea of the light cone
and "spacelike" and "timelike" separations between two events. Con
sider Fig. 1 1 .10, in which the time axis (actually ct) is vertical and the space
axes are perpendicular to it. For simplicity only one space dimension is
shown. At t = a physical system, say a particle, is at the origin. Because
the velocity of light is an upper bound on all velocities, the spacetime
domain can be divided into three regions by a "cone," called the light cone,
whose surface is specified by x 2 + y 2 + z 2 = c 2 t 2 . Light signals emitted
at / = from the origin would travel out the 45° lines in the figure. But
any material system has a velocity less than c. Consequently as time goes
on it would trace out a path, called its world line, inside the upper half
cone, e.g., the curve OB. Since the path of the system lies inside the upper
halfcone for times t > 0, that region is called the future. Similarly the
lower halfcone is called the past. The system may have reached O by a
path such as AO lying inside the lower halfcone. The shaded region
outside the light cone is called elsewhere. A system at O can never reach
or come from a point in spacetime in elsewhere.
The division of spacetime into the pastfuture region and the elsewhere
region can be emphasized by considering the invariant separation between
two events P x {x x , y x , z x , t x ) and P 2 (x 2 , y 2 > z 2> h) i n spacetime :
*i2 2 = (*i  * 2 ) 2 + G/i  y 2 f + (*i  * 2 ) 2  c\t x  t 2 ) 2 (1 1.64)
[Sect. 11.7] Special Theory of Relativity 371
For any two events P x and P 2 there are two possibilities : (1) s n 2 > 0,
(2) s 1% 2 < 0. If 5 12 2 > 0, the events are said to have a spacelike separation,
because it is always possible to find a Lorentz transformation to a new
coordinate system K' where (// — t 2 ') = and
* 12 2 = «  x 2 'f + (<//  y 2 'f + &'  z 2 ') 2 > (1 1.65)
That is, the two events are at different space points at the same instant of
time. In terms of Fig. 11.10, one of the events is at the origin and the other
lies in elsewhere. If s 12 2 < 0, the events are said to have a timelike separ
ation. Then a Lorentz transformation can be found which will make x{
= *2, Vi = Vz, z i = z 2> and
V= c\t{  t 2 'f < (11.66)
In the coordinate system K' the two events are at the" same space point,
but are separated in time. In Fig. 11.10, one point is at the origin and the
other is in the past or future.
The division of the separation of two events in spacetime into two
classes — spacelike separations or timelike separations — is a Lorentz
invariant one. Two events with a spacelike separation in one coordinate
system have a spacelike separation in all coordinate systems. This means
that two such events cannot be causally connected. Since physical inter
actions propagate from one point to another with velocities no greater
than that of light, only events with timelike separations can be causally
related. An event at the origin in Fig. 11.10 can be influenced causally
only by the events which occur in the past region of the light cone.
11.7 Lorentz Transformations as Orthogonal Transformations in
Four Dimensions
The Lorentz transformation (11.19) and the more general form (11.21)
are linear relations between the spacetime coordinates (x, y, z, i) and
(x', y', z', t'\ subject to the constraint,
x 2 __ y 2 + 2 2 _ c 2,2 = x >2 + y >2 + z >2 _ fr'2 (H.67)
This constraint is very reminiscent of the constraint involved in the
rotation of coordinate axes in three space dimensions. In fact, if we intro
duce the four spacetime coordinates,
x ± = x, x 2 = y, x 3 = z, # 4 = ict (11.68)
then the constraint becomes
F 2 = x* + V + * 3 2 + x? (1 1 .69)
372 Classical Electrodynamics
is an invariant under Lorentz transformations. This is then exactly the
requirement that Lorentz transformations are rotations in a fourdimen
sional Euclidean space or, more correctly, are orthogonal transformations
in four dimensions. The Lorentz transformation (11.21) can be written
in the general form :
V=IVv, /* =1,2, 3, 4 (11.70)
v = l
where the coefficients a^ v are constants characteristic of the particular
transformation. The invariance of R 2 (11.69) forces the transformation
coefficients a^ to satisfy the orthogonality condition :
4
2 fl MV^A = ^vA (1171)
« = 1
With (11.71) it is easy to show that the inverse transformation is
4
x n = 2 <«v„ (11.72)
v = l
and that .
4
2 a v M fl ^ = <5vA (11.73)
H = l
Furthermore, if we solve the four equations (11.70) for x^ in terms of xj
and compare the solution to (11.72), we find that the determinant of the
coefficients has the value unity :
det a„ v  = 1 (11.74)
In general the determinant can be ±1, but the choice of the minus sign
implies an inversion followed by a rotation.
To give some substance to the above formalities we exhibit explicitly
the transformation coefficients a^ v for a Lorentz transformation from
system K to a system K' moving with a velocity v parallel to the z axis
'l
10
^) = I n J (H75)
y iyfi
iO —iyfl y
We have introduced the convenient abbreviations :
c
1
y =
i4
c 2
(11.76)
[Sect. 11.7]
Special Theory of Relativity
373
Fig. 11.11
Lorentz transformation as
rotation of axes.
With definitions (11.68) and (11.70) it is elementary to show that (11.75)
yields exactly the Lorentz transformation (11.19).
The formal representation of transformation (1 1.75) as a rotation of axes
in the x 3 , x± plane (with a; 4 drawn as if it were real) can be accomplished
simply. Figure 11.11 shows a rotation of the axes through an angle ip . The
coordinates of the point P relative to the two sets of axes are related by
x 3 = cos ip x 3 + sin ip x±
xl = —sin ip x 3 + cos ip £ 4
(11.77)
Comparison of the coefficients in (11.77) with the transformation coeffi
cients in (11.75) shows that the angle ip is a complex angle whose tangent
is
tanip = ip (11.78)
This result can be obtained directly from (11.77) without reference to
(11.75) by noting that the origin x 3 = moves with a velocity v in the
system K. That the angle ip is complex is emphasized by the fact that its
cosine is greater than unity (cos ip = y > 1). Consequently the graphical
representation of a Lorentz transformation as a rotation is merely a formal
device.
In spite of the formal nature of the x 3 , # 4 rotation diagram the pheno
mena of FitzGeraldLorentz contraction and time dilatation can be
displayed graphically. Figure 11.12 shows the length contraction on the
right and time dilatation on the left. The distance L in the frame K' is
observed in the frame AT as L, represented by the horizontal line at constant
time in K. Because of the complex nature of the angle ip, L appears on the
figure as larger than L , but mathematically the two lengths are related by
or
Lcos ip = L
7 J
(11.79)
Classical Electrodynamics
Fig. 11.12 Time dilatation and Fitz
GeraldLorentz contraction in terms
of a rotation of spacetime axes.
in agreement with (11.22). Similarly the time intervals T in the frame K'
are seen in the frame K as intervals T, where
T = T cos y> = yT (1 1.80)
in accord with (11.24).
Sometimes a graphical display of Lorentz transformations is made
using a real time variable x = ct, rather than ar 4 . This is called a
Minkowski diagram and has the virtue of dealing with real quantities. It
has the major disadvantage that the coordinate grids in the two frames K
and K' must be scaled according to a rectangular hyperboloid law, as can
be seen from (11.67). The interested reader may refer to Minkowski's
paper in the collection, The Principle of Relativity, by Einstein et al., for a
discussion of these diagrams.
11.8 4Vectors and Tensors; Covariance of the Equations of Physics
The transformation law (11.70) for the coordinates x^ defines the trans
formation properties of vectors in the fourdimensional spacetime (1 1.68).
Any set of four quantities A^ which transform in the same way as x^ is
called a 4vector. Under the Lorentz transformation (a^) A^is transformed
into A' where
Ap — 2, a nv^\
(11.81)
If a quantity <f> is unchanged under a Lorentz transformation, it is called
a scalar or a Lorentz scalar. The four quantities formed by differentiation
of a Lorentz scalar with respect to x^ transform as a 4vector. This can be
shown as follows. Consider
d<f> _^d<f> dx v
dx' J* dx v dx'
* v = l M
(11.82)
[Sect. 11.8] Special Theory of Relativity 375
From (11.72) it is evident that
^7 = «„ (1183)
Consequently
^^%*4 (ii 84 )
ox v
dx' *i
as required for the transformation of a 4vector. By similar means it is
elementary to show that the 4divergence of a 4vector is Lorentz invariant :
ydAl = y<^ (1L85)
With A = dcfyjdx^ in this expression, we find that the fourdimensional
Laplacian operator is a Lorentz invariant operator :
^ifMg** (1L86)
v=l V n=l *
If D 2 operates on some other function, such as a 4vector A^ the resulting
quantity retains the transformation properties of the function operated
on. The scalar product of two 4vectors A^ and B^ is readily proved to be
invariant :
(i4'B')s2^'B/ = (i4B) (1187)
H = l
Lorentz 4vectors are tensors of the first rank in a fourdimensional
space. Higherranks tensors are defined in an analogous way. A second
rank tensor T is a set of sixteen quantities which transforms according
to the law:
T; v = I a^T* (11.88)
A, (7 = 1
Higherrank tensors are formed by the inclusion of more and more factors
a v . A tensor of the «th rank is a set of 4 n quantities which have a transfor
mation law involving a product of n coefficients a^, in obvious generali
zation of (1 1 .88). Just as the scalar product of two 4vectors has rank one
less than the original quantities, so certain contracted quantities can be
formed from higherrank tensors. For example, the scalar product of a
tensor of the second rank and a 4vector transforms as a 4vector:
B; = I T^A\ = ZaJ2 T, V A V ) (11.89)
v = l A = l \v = l /
This and similar relations can be proved using the orthogonality con
ditions (11.71) and (11.73).
376 Classical Electrodynamics
The volume element in the fourdimensional spacetime (11.68) will be
defined as the real quantity
d*x = dx x dx 2 dx 3 dx (11.90)
where dx Q = (l/i) dx^ = d{ci). The transformation law of the volume
element is
d 4 x' = d«> X *> X 3> «*') fa (U 91)
0\X^ x^, x 3 , X^)
But the Jacobian in (11.91) is just the determinant of the a (11.74).
Consequently the 4 volume element d*x is a Lorentz invariant quantity.
The first principle of Einstein is that the laws of physics have the same
form in different Lorentz frames. This means that the equations which we
write down to describe the physical laws must be covariant in form. By
covariant we mean that the equation can be written so that both sides have
the same, welldefined, transformation properties under Lorentz transfor
mations. Thus physical equations must be relations between 4vectors, or
Lorentz scalars, or in general 4tensors of the same rank. This is necessary
in order that a relation valid in one coordinate frame will also hold in the
same form in another. Consider, for example, the inhomogeneous pair of
Maxwell's equations. It will be shown in the next section that these can be
written in the relativistic form
\SdF 4tt
yr 1 = J^ ?* = 1,2, 3, 4 (11.92)
v tt dx * c
where J ^ is a suitable current 4 vector, and F^ is the field strength 4tensor.
Since the 4divergence of a 4tensor is a 4 vector, (11.92) is a relation
between two 4vectors. In another reference frame K', we expect the same
physical laws to take the same form,
tf^' = ^ (1193)
^J ox a c
Using transformation (11.81), we find that (11.93) can be expressed in
terms of quantities in the original coordinate frame as
i>(i:2:T'')
fl = l v = l v
This shows that, if (1 1.92) holds in the original frame of reference, then it
holds in all equivalent Lorentz frames. If the two sides of (1 1.92) had not
had the same Lorentz transformation properties, this would obviously
not be true.
[Sect. 11.9] Special Theory of Relativity 377
To conclude these formal considerations we introduce some simplifying
notation. In what follows :
1. Greek indices will be summed from 1 to 4.
2. Roman indices will represent spatial directions and will be summed
from 1 to 3.
3. 4 vectors will be denoted by A^ with (A lt A 2 , A 3 ) the components of
a space vector A and A^ = iA . This correspondence will sometimes be
written
A„ = (A,iA ) (11.95)
Sometimes the subscript on the 4vector will be omitted, e.g. f(x) means
/(x, 0
4. Scalar products of 4 vectors will be denoted by
(AB) = A B A B (11.96)
where A • B is the ordinary 3space scalar product.
5. The summation convention will be used. That is, repeated indices
are understood to be summed over, even though the summation sign is not
written. If the repeated index is roman, the sum is from 1 to 3 ; if it is
Greek, the sum is from 1 to 4. Thus, for example, (11.85) will be written
dA v ' = BA,
dx v ' dx^
and (11.89) will be written
Tfi V A v = a llX T }v A v
11.9 Covariance of Electrodynamics
The invariance in form of the equations of electrodynamics under
Lorentz transformations was shown by Lorentz and Poincare before
Einstein formulated the special theory of relativity. We will now discuss
this covariance and consider its consequences. There are two points of
view possible. One is to take some experimentally proven fact such as the
invariance of electric charge and try to deduce that the equations must be
covariant. The other is to demand that the equations be covariant in form
and to show that the transformation properties of the various physical
quantities, such as field strengths and charge and current, can be satis
factorily chosen to accomplish this. Although the first view is to some the
most satisfying, we will adopt the second course. Classical electrodynamics
is correct, and it can be cast in covariant form. For simplicity we will
consider the microscopic equations, without the derived quantities D
andH.
378 Classical Electrodynamics
We begin with the continuity equation for charge and current densities :
^=V.J (11.97)
dt
This can be cast in covariant form by introducing the chargecurrent 4
vector Jp defined by
J„ = (J, icp) (11.98)
Then (11.97) takes on the obviously covariant form:
^ = (11.99)
That J^ is a legitimate 4vector can be established from the experimentally
known invariance of electric charge. This invariance implies that
(p dx x dx 2 dx z ) is a Lorentz invariant. Since i d*x = (dx x dx 2 dx 3 dx^ is a
Lorentz invariant, it follows that p transforms like the fourth component of
a 4vector. The transformation properties of J follow similarly.
The wave equations for the vector potential A and the scalar potential
O are
c 2 dt 2 c
V 2 0— = 4tt P
with the Lorentz condition
(11.100)
VA + i— = (11.101)
c dt
The differential operator on the lefthand sides of the wave equations can
be recognized as the Lorentz invariant fourdimensional Laplacian (1 1.86).
The righthand sides of these equations are the components of a 4vector.
Consequently, the requirement of covariance means that the vector and
scalar potentials are the space and time parts of a 4vector potential A :
4, = (A,iO) (11.102)
Then the wave equations can be written
□ 2 ^ =  ^L j ju = 1,2, 3, 4 (11.103)
c
while the Lorentz condition becomes
^ = (11.104)
dx^
[Sect. 11.9]
Special Theory of Relativity
379
We are now ready to consider the field strengths E and B. They are
defined in terms of the potentials by
c dt
B = V x A J
By writing out the components explicitly, for example,
(11.105)
dA x dA±
ltl = a a
ox± ox 1
B = dA 3 dA 2
(11.106)
it is evident that the electric field and the magnetic induction are elements
of the secondrank, antisymmetric, fieldstrength tensor F^:
(11.107)
P _2±._
dA_,
dx v
Explicitly, the fieldstrength tensor is
/ B z
B 2
— IE^
lB 3
*i
— iE 2
(iV) = 1 B ^ _ Bi
iE z
\ iE x iE%
iE 3
(11.108)
To complete the demonstration of the covariance of electrodynamics
we must consider Maxwell's equations. The inhomogeneous pair are
V • E = 4rrp
c dt c
(11.109)
Since the righthand sides form the components of a 4 vector, so must the
lefthand sides. With definition (11.108) of the fieldstrength tensor it is
easy to show that the lefthand sides in (11.109) are the divergence of the
fieldstrength tensor. Thus (11.109) takes the covariant form
wl JIV ^" r
dx v c
Similarly the two homogeneous Maxwell's equations,
1 f5R
V • B = 0, VxE + ~=0
c dt
(11.110)
(11.111)
380 Classical Electrodynamics
can be shown to reduce to the four equations :
dF„ v dF Xll dF vX
— ^ + ^ + ^ = (11.112)
ox x dx v dXp
where A, ju, v are any three of the integers 1 , 2, 3, 4. Each term in (1 1 . 1 12)
transforms like a 4tensor of the third rank so that the equation is covariant
in form, as required.
11.10 Transformation of the Electromagnetic Fields
Since the fields E and B are elements of the fieldstrength tensor F ,
their transformation properties can be found from
F'.y = a^a^F^ (11.113)
With transformation (11.75) from a system Kto K' moving with velocity
v along the x 3 axis, (11.113) gives the transformed fields:
E{ = 7(E X  PB 2 ) 2?/ = y{B 1 + 0EJ
E* = y(E 2 + PBJ B 2 ' = y(B 2  $E X )
Ez — E 3 B 3 = B 3
(11.114)
The inverse transformation can be obtained from (11.114) by the inter
change of primed and unprimed quantities and (5 > — /?. For a general
Lorentz transformation from K to a system K' moving with velocity v
relative to K, the transformation of the fields is evidently
E M ' = E M B„ ' = B M
E/ = y( El +  c x b) B/ = y ( Bl  1 x E ) J (1L115)
Here  and j^ mean parallel and perpendicular to the velocity v. Transfor
mation (11.115) shows that E and B have no independent existence. A
purely electric or magnetic field in one coordinate system will appear as a
mixture of electric and magnetic fields in another coordinate frame. Of
course certain restrictions apply (see Problem 11.10) so that, for example,
a purely electrostatic field in one coordinate system cannot be transformed
into a purely magnetostatic field in another. But the fields are completely
interrelated, and one should properly speak of the electromagnetic field
F^, rather than E or B separately.
As an example of the transformation of the electromagnetic fields we
consider the fields seen by an observer in the system K when a point charge
q moves by in a straightline path with a velocity v. The charge is at rest
in the system K', and the transformation of the fields is given by the inverse
[Sect. 11.10]
Special Theory of Relativity
381
Fig. 11.13 Particle of charge q
moving at constant velocity v
passes an observation point P
at impact parameter b.
X\
^\ r
X\
a v
i > —
X3
9
*3
*2
'X%
of (1 1.1 14) or (11.115). We suppose that the charge moves in the positive
x 3 direction and that its closest distance of approach to the observer is b.
Figure 11.13 shows a suitably chosen set of axes. The observer is at the
point P. At / = t' = the origins of the two coordinate systems coincide
and the charge q is at its closest distance to the observer. In the frame K'
the observer's point P, where the fields are to be evaluated , has coord inates
Xl ' = b, x 2 ' = 0, < = — vt', and is a distance r' = Vb 2 + {vt'f away
from*?. We will need to express r' in terms of the coordinates of K. The
only coordinate needing transformation is the time t ' = y[t — (vjc 2 )x 3 ] =
yt, since x 3 = for the point P in the frame K. In the rest frame K' of the
charge the electric and magnetic fields are
£/ = 0,
£,' = 0,
B 2 ' = 0,
E,' =
BJ =
qvt'
~'3
(11.116)
In terms of the coordinates of K the nonvanishing field components are
£/ =
qb
£,'= 
qyvt
(11.117)
(6 2 + yW? A ' ~* (b 2 + y W)° A
Then, using the inverse of (11.114), we find the transformed fields in the
system K:
yqb
E x = yE{ =
£o = £q = —
(&*.+ yW) 8/ *
qyvt
(b 2 + yWf*
B 2 = y^ = fiEi
(11.118)
with the other components vanishing.
Fields (11.118) exhibit interesting behavior when the velocity of the
charge approaches that of light. First of all there is observed a magnetic
382
Classical Electrodynamics
induction in the x 2 direction. This magnetic field becomes almost equal to
the transverse electric field E x as /8 > 1. Even at nonrelativistic velocities
where y c=L 1, this magnetic induction is equivalent to
B
q v x r
(11.119)
which is just the AmpereBiotSavart expression for the magnetic field of
a moving charge. This can obviously be obtained directly from the
inverse of (11.115). At high speeds when y > 1 we see that the peak
transverse electric field E x (t = 0) becomes equal to y times its nonrelati
vistic value. In the same limit, however, the duration of appreciable field
strengths at the point P is decreased. A measure of the time interval over
which the fields are appreciable is evidently
Lt~±
b_
yv
(11.120)
As y increases, the peak fields increase in proportion, but their duration
goes in inverse proportion. The time integral of the fields times v is
independent of velocity. Figure 11.14 shows this behavior of the transverse
electric and magnetic fields and the longitudinal electric field. For /S » 1
the observer at P sees nearly equal transverse and mutually perpendicular
electric and magnetic fields. These are indistinguishable from the fields
of a pulse of plane polarized radiation propagating in the x z direction.
79
b 2
P<1
W
L_itI
Is"
***v\
vt >
Fig. 11.14 Fields due to a uniformly moving, charged particle as a function of time.
[Sect. 11.11] Special Theory of Relativity 383
The extra longitudinal electric field varies rapidly from positive to
negative and has zero time integral. If the observer's detecting apparatus
has any significant inertia, it will not respond to this longitudinal field.
Consequently for practical purposes he will see only the transverse fields.
This equivalence of the fields of a relativistic charged particle and those of
a pulse of electromagnetic radiation will be exploited later in Chapter 1 5.
That a plane electromagnetic wave in one coordinate frame K will also
appear as a plane wave in another coordinate frame K' moving with
constant velocity relative to K follows from the invariant form of the wave
equation under Lorentz transformations. Thus in the frame K a plane
wave is represented by
^v(x,0=/,/ k  x ^ (11121)
where f^ are appropriate constant coefficients, and k and co are the wave
vector and frequency of the wave. In the coordinates system K' the plane
will be
i 7 ;v(x',o=/;v^ k ' x '~ icor (n.122)
where the f'^ are again constant coefficients, and k' and co' are the wave
vector and frequency as seen in K'. According to (1 1.1 13), the two sets of
fields are related by
ri ik' .x' —i<o't' r ik.x—ia>t /i i n ^
//•v e = a&<*v*fjL*e (11.123)
In order that (11.123) be true at all points in spacetime the phase factors
on both sides must be equal:
k'x' — co't' = k.x  cot (11.124)
This invariance of the phase means that k and co must form the space and
time parts of a 4 vector k^:
^('•t)
(11.125)
Then the invariance of phase becomes the obvious invariance of a scalar
product (k • x) of two 4vectors. The relativistic formulas for the Doppler
shift follow immediately from (11.125), as was shown in Section 11.4.
11.11 Covariance of the Force Equation and the Conservation Laws
In Section 11.9 the covariance of electrodynamics was discussed from
the point of view of charge and current densities and the resulting fields
and potentials. We know that the sources of charge and current are
ultimately charged particles which can move under the action of fields.
384 Classical Electrodynamics
Consequently to complete our discussion we must consider the covariant
formulation of the Lorentz force equation and the conservation laws of
momentum and energy.
The Lorentz force equation can be written as a force per unit volume
(representing the rate of change of mechanical momentum of the sources
per unit volume) :
f=pE + J xB (11.126)
c
where J and p are the current and charge densities. Writing out a single
component of f, we find
A = P E X +  (J 2 B 3  J 3 B 2 ) =  (F 12 J 2 + F 13 J 3 + F 14 J 4 ) (11.127)
c c
where we have used definitions (1 1 .98) and (1 1 .108). The other components
of f yield similar results, showing that (11.126) can be written as
f k = F kv J v , k= 1,2,3 (11.128)
c
The righthand side of (11.128) is evidently the space components of a
4 vector. Hence f must be the space part of a 4 vector /„ = If, i — I,
where :
/„ = V* ( 11129 >
c
To see the meaning of the fourth component of the forcedensity 4vector
we write out
f = ~h =  {F^i + V2 + iVs) = E • J (11.130)
i i
But (E • J) is just the rate at which the field does work on the sources per
unit volume, or the rate of change of mechanical energy of the sources per
unit volume. Thus we see that the covariant form (1 1.129) of the Lorentz
force equation gives the rate of change of mechanical momentum per unit
volume as its space part, and the rate of change of mechanical energy per
unit volume as its time part. Alternatively, it may be viewed as giving the
space and time derivatives of something of the dimensions of work per
unit volume.
The conservation laws for mechanical plus electromagnetic energy and
momentum derived in Chapter 6 can be presented in covariant form as
the space and time components of a single 4vector equation. If the in
homogeneous Maxwell's equations (11.110) are used to eliminate J v in
[Sect. 11.11] Special Theory of Relativity
(11.129), the force density becomes
F„
J II . * /JV
47T
1  9F V ,
dx }
385
(11.131)
The righthand side of (1 1.131) can be written as the divergence of a tensor
of the second rank. We define the symmetric tensor T^, called the
electromagnetic stressenergymomentum tensor,
T L
<terL
FnxFxv + i^u V Fi a F_
fiX* Av
? u ttv r A<r r A<r
(11.132)
It will be left to the problems (Problem 11.12) to show that by means of
the homogeneous Maxwell's equations and (1 1.132) force equation (11.131)
can be written in the form:
/„ = ^ (H133)
ox v
The tensor T^ v can be written out explicitly in terms of the fields using
(11.132):
'll 119 lift
<T„) =
J 21
T 2 2 T 23
^32 *33
icg 2 icg 3
Kgl)
icgz
icgz
u
(11.134)
where T ik is the symmetric Maxwell's stress tensor defined on page 194,
g is the electromagnetic momentum density,
g = — E xB = ^S
Aire c 2
and u is the energy density,
u = — (E 2 + B 2 )
(11.135)
From definition (6.102) of the spatial parts of T^ [or from (11.132)], we
see that the stressenergymomentum tensor has a vanishing trace :
IT,, = (11.136)
The conservation laws of momentum and energy are merely the three
dimensional integrals of the force equation (1 1.133). To see this we write
out a typical spatialcomponent equation :
Jk —
dr t
dx„
dx, dx.
v A kv w l ki i w *M /T7 . "T \ "gk
dt
(11.137)
386 Classical Electrodynamics
If we identify the spatial integral of f k as the rate of change of the kth
component of mechanical momentum P k , then the integral of (11.137) can
be written
4 (P + G) = I V •¥ d z x = i> n V da (11.138)
dt Jv Js
where G k is the kth component of total electromagnetic momentum. This
is the momentumconservation law already obtained in Chapter 6.
Similarly the fourth component of (11.133) can be written
/ = £/ 4 = E .J = ^ + ^= VS^ (11.139)
i i ox i i ox± dt
With the volume integral of f identified as the rate of change of total
mechanical energy T, the conservation of energy law is
{T+ U)=  I VSd 3 x =  <t> nS da (11.140)
dt Jv Js
where U is the total electromagnetic energy in the volume V.
REFERENCES AND SUGGESTED READING
The theory of relativity has an extensive literature all its own. To my mind the most
lucid, though concise, presentation of special and general relativity is the famous 1921
article (recently brought up to date) by
Pauli.
In addition, there are a number of textbooks devoted to special relativity at the graduate
level, some of which are
Aharoni,
Bergmann, Chapters IIX,
Moller, Chapters IVII.
Moller's book is perhaps the most authoritative.
The flavor of the original theoretical developments can be obtained by consulting
the collected papers of
Einstein, Lorentz, Minkowski, and Weyl.
The main experiments are summarized briefly, but clearly, in
Moller, Chapter I,
Panofsky and Phillips, Chapter 14.
A fuller description of the experimental basis of special relativity, with many references,
is presented in
Condon and Odishaw, Part 6, Chapter 8 by E. L. Hill.
Thomas precession is discussed by
Moller, Sections 22 and 47,
in a manner akin to ours. A different approach to the problem is given by
Corben and Stehle, Section 92.
[Probs. 11]
Special Theory of Relativity
387
PROBLEMS
11.1 A possible clock is shown in the figure. It consists of a flashtube F and a
photocell P shielded so that each views only the mirror M, located a
distance d away, and mounted rigidly with respect to the flashtubephoto
cell assembly. The electronic innards of the box are such that, when the
photocell responds to a light flash from the mirror, the flashtube is
triggered with a negligible delay and emits a short flash towards the
mirror. The clock thus "ticks" once every (2d/c) seconds when at rest.
11.2
(a) Suppose that the clock moves with a uniform velocity v, perpen
dicular to the line from PF to M, relative to an observer. Using the
second postulate of relativity, show by explicit geometrical or algebraic
construction that the observer sees the relativistic time dilatation as the
clock moves by.
(Z>) Suppose that the clock moves with a velocity v parallel to the line
from PF to M. Verify that here, too, the clock is observed to tick more
slowly, by the same time dilatation factor.
(a) Show explicitly that two successive Lorentz transformations in the
same direction commute and that they are equivalent to a single Lorentz
transformation with a velocity
v x + v 2
1 + Ov 2 /c 2 )
11.3
This is an alternative way to derive the parallelvelocity addition law.
(b) Show explicitly that two successive Lorentz transformations at right
angles (y x in the x direction, v % in the y direction) do not commute. Show
further that in whatever order they are applied the result is not the same
as a single transformation with y =w x + jv 2 . Give one or more simple
reasons why this result is necessary within the framework of special
relativity.
(a) Find the form of the wave equation in system A" if it has its standard
388 Classical Electrodynamics
form in system K' and the two coordinate systems are related by the
Galilean transformation x' = x — vt, t' = t.
(b) Show explicitly that the form of the wave equation is the same in
system K as in K' if the coordinates are related by the Lorentz trans
formation x' = y(x — vt), t' = y[t — (vx/c 2 )].
11.4 A coordinate system K' moves with a velocity v relative to another system
K. In K' a particle has a velocity u' and an acceleration a'. Find the
Lorentz transformation law for accelerations, and show that in the system
K the components of acceleration parallel and perpendicular to v are
9,,= FW a "
llt\
\ c 2 } ( v \
a ±=7 X3 W +  2 X < a ' Xu)
11.5 Assume that a rocket ship leaves the earth in the year 2000. One of a set
of twins born in 1980 remains on earth; the other rides in the rocket.
The rocket ship is so constructed that it has an acceleration g in its own
rest frame (this makes the occupants feel at home). It accelerates in a
straightline path for 5 years (by its own clocks), decelerates at the same
rate for 5 more years, turns around, accelerates for 5 years, decelerates for
5 years, and lands on earth. The twin in the rocket is 40 years old.
(a) What year is it on earth?
(b) How far away from the earth did the rocket ship travel ?
11.6 In the reference frame K two very evenly matched sprinters are lined up a
distance d apart on the y axis for a race parallel to the x axis. Two
starters, one beside each man, will fire their starting pistols at slightly
different times, giving a handicap to the better of the two runners. The
time difference in K is T.
(a) For what range of time differences will there be a reference frame K'
in which there is no handicap, and for what range of time differences is
there a frame K' in which there is a true (not apparent) handicap?
(b) Determine explicitly the Lorentz transformation to the frame K'
appropriate for each of the two possibilities in (a), finding the velocity of
K' relative to K and the spacetime positions of each sprinter in K'.
11.7 Using the fourdimensional form of Green's theorem, solve the inhomo
geneous wave equations
— 4n
(a) Show that for a localized chargecurrent distribution the 4vector
potential is
A(x) = I fep d^
" 7TC J R 2
[Probs. 11] Special Theory of Relativity 389
where R 2 = (x — f) • (x — ), x means (x lt x 2 ,x 3 , # 4 ), and J 4  = d£ x d£ 2 d£ 3 d 4 .
(b) From the definitions of the field strengths F^ show that
= _2 (*(/ x
TTCj R
/*" ~ I — 7j3 — s
where (7 x i?)^ = /^/f,  J V R^
11.8 The threedimensional formulation of the radiation problem leads to the
retarded solution
A^t)^lj J J^l2
(PS
t'=t(rje)
where r = x — %\. Show the connection between this retarded solution
and the solution of Problem 11.7 by explicitly performing the integration
over d£ 4 .
11.9 A classical point magnetic moment f* at rest has a vector potential
and no scalar potential. Show that, if the magnetic moment moves with a
velocity v(v <^ c), there is an electric dipole moment p associated with the
magnetic moment, where
v
p = X {X
c
What can you say if v is not small in magnitude compared toe? Show
that the interaction energy between the moving dipole and fields E and B
is the same as would be obtained by calculating the magnetic field in the
rest frame of the magnetic moment.
11.10 (a) Show that (B 2 — E 2 ) is an invariant quantity under Lorentz trans
formations. What is its form in fourdimensional notation ?
(b) The symbol e Xflva is defined to have the properties
ro if
if any two indices are equal
for an even (odd) permutation of indices
€ Xllva is a completely antisymmetric unit tensor of the fourth rank (actually
a pseudotensor under spatial inversion). Prove that ^XfivaF^va (sum
mation convention implied) is a Lorentz invariant, and find its form in
terms of E and B.
11.11 In a certain reference frame a static, uniform, electric field E is parallel to
the x axis, and a static, uniform, magnetic induction B = 2E lies in the
xy plane, making an angle 6 with the x axis. Determine the relative
velocity of a reference frame in which the electric and magnetic fields are
parallel. What are the fields in that frame for << 1 and »> (tt/2) ?
390 Classical Electrodynamics
11.12 Show that the force equation/,, = (l/c)F MV J v can be written as
dT
U dx v
where
T l*v = ^\. F nX F X V + i S nv F XaFxo\
11.13 A pulse of electromagnetic radiation of finite spatial extent exists in charge
and currentfree space.
(a) By means of the divergence theorem in four dimensions, prove that
the total electromagnetic momentum and energy transform like a 4 vector.
(b) Show that for a plane wave this 4 vector has zero "length," but
that for other possible field configurations (e.g., spherically diverging wave)
this is not true.
12
RelativisticParticle
Kinematics and Dynamics
In Chapter 1 1 the special theory of relativity was developed with
particular emphasis on the electromagnetic fields and the covariance of
the equations of electrodynamics. Only in Section 11.11 was there a
mention of the mechanical origin of the sources of charge and current
density. The emphasis on electromagnetic fields is fully justified in the
presentation of the first aspects of relativity, since it was the behavior of
light which provided the puzzling phenomena that were understood in
terms of the special theory of relativity. Furthermore, a large class of
problems can be handled without inquiry into the detailed mechanical
behavior of the sources of charge and current. Nevertheless, problems
which emphasize the fields rather than the sources form only a part of
electrodynamic phenomena. There is the converse type of problem in
which we are interested in the behavior of charged particles under the
action of applied electromagnetic fields. The particles represent charge
and current densities, of course, and so act as sources of new fields. But
for most applications these fields can be neglected or taken into account in
an approximate way. In the present chapter we wish to explore the motion
of relativistic particles, first their kinematics and then their dynamics in
external fields. Discussion of the difficult problem of charged particles
acting as the sources of fields and being acted on by those same fields will
be deferred to Chapter 17.
12.1 Momentum and Energy of a Particle
In nonrelativistic mechanics a particle of mass m and velocity v has a
momentum p = rav and a kinetic energy T = \mv^. Newton's equation
391
392 Classical Electrodynamics
of motion relates the time rate of change of momentum to the applied
force. For a charged particle the force is the Lorentz force. Since we have
discussed the Lorentz transformation properties of the Lorentz force density
in Section 11.11, we can immediately deduce the behavior of a charged
particle's momentum under Lorentz transformations. For neutral particles
with no detectable electromagnetic interactions it is clearly impossible to
obtain their relativistic transformation properties in this way, but there
is ample experimental evidence that all particles behave kinematically in
the same way, whether charged or neutral.
A charged particle can be thought of as a very localized distribution of
charge and mass. To find the force acting on such a particle we integrate
the Lorentz force density/^ (11.129) over the volume of the charge. If the
total charge is e and the velocity of the particle is v, then the volume
integral of (11. 129) is
f tl d 3 x = e F MV v v (12.1)
c
S
where v„ = (v, ic), and F^ v is interpreted as the average field acting on the
particle. The lefthand side of (12.1) is now to be equated to the time rate
of change of the momentum and energy of the particle, just as in Section
11.11. Thus
^ //„«*•* (12.2)
where we have written p k as the kXh component of the particle's momentum
and /> 4 = iEjc as proportional to the particle's energy. That p^ is indeed
a 4 vector follows immediately from (12.2). If we integrate both sides with
respect to time, then the lefthand side becomes the momentum or energy
of the particle while the righthand side is the fourdimensional integral
of f^. Since d*x is a Lorentz invariant quantity, it follows that p^ must
have the same transformation properties as/„. Therefore the momentum
p and the energy E of a particle form a 4 vector p^ :
Pll =U) (123)
" v c
The transformation of momentum and energy from one Lorentz frame K
to another K' moving with a velocity v parallel to the z axis is
Pi = Pi> Pz = Pz
p s = y \Ps + P —J
E = y(E' + jfcfl,')
(12.4)
[Sect. 12.1] RelativisticF 'article Kinematics and Dynamics 393
where /? = vjc and y = (1 — /S 2 )  ^. The inverse transformation is
obtained by changing /5 — »► — /5 and interchanging the primed and unprimed
variables.
The length of the 4 vector p^ is a Lorentz invariant quantity which is
characteristic of the particle :
(P'P) = (p'p')= 2 (12.5)
c 2
In the rest frame of the particle (p' = 0) the scalar product (12.5) gives the
energy of the particle at rest :
E' = X (12.6)
To determine X we consider the Lorentz transformation (12.4) of p^ from
the rest frame of the particle to the frame Kin which the particle is moving
in the z direction with a velocity v. Then the momentum and energy are
*>#
(12.7)
E = yX
From the nonrelativistic expression for momentum p = my we find that
the invariant constant X = mc 2 . The nonrelativistic limit of the energy is
E = ymc 2 ~ mc 2 + \mv 2 \ (12.8)
This shows that is is the total energy of the particle, consisting of two parts :
the rest energy (mc 2 ) and the kinetic energy. Even for a relativistic particle
we can speak of the kinetic energy T, defined as the difference between the
total and the rest energies :
T = E  mc 2 = (y  \)mc 2 (12.9)
In summary, a free particle with mass m moving with a velocity v in a
reference frame K has a momentum and energy in that frame :
p = yms )
E = ymc* J
From (12.5) it is evident that the energy E can be expressed in terms of the
momentum as
E = (c 2 p 2 + m 2 c*) v * (12.11)
The velocity of the particle can likewise be expressed in terms of its
momentum and energy :
v = ^ (12.12)
E
394
Classical Electrodynamics
In dealing with relativisticparticle kinematics it is convenient to adopt
a consistent, simple notation and set of units in which to express momenta
and energies. In the formulas above we see that the velocity of light appears
often. To suppress various powers of c and so simplify the notation we
will adopt the convention that all momenta, energies, and masses will be
measured in energy units, while velocities are measured in units of the
velocity of light. All powers of c will be suppressed. Consequently in
what follows, the symbols
P
E
m
v
stand for
cp
E
v/c
(12.13)
As energy units, the ev (electron volt), the Mev (million electron volt), and
the Bev (10 9 ev) are convenient. One electron volt is the energy gained by
a particle with electronic charge when it falls through a potential difference
of one volt (1 ev = 1.602 x 10" 12 erg).
12.2 Kinematics of Decay Products of an Unstable Particle
As a first illustration of relativistic kinematics which follow immediately
from the 4 vector character of the momentum and energy of particles, we
consider the twobody decay of an unstable particle at rest. Such decay
processes are common among the unstable particles. Some examples are
the following.
1. Charged pi meson decays into a mu meson and a neutrino with a
lifetime t = 2.6 x 10~ 8 sec:
77" — *■ jU + V
The pimeson rest energy is M = 139.6 Mev, while that of the mu meson
is m^ = 105.7 Mev. The neutrino has zero rest mass, m v = 0. There is,
therefore, an energy release of 33.9 Mev in pimeson decay.
2. Charged K meson sometimes decays into two pi mesons with a
lifetime t = 1.2 x 10 8 sec:
K±
+ 77°
The charged K meson has a rest energy M = 494 Mev, while the two pi
mesons have rest energies, m ± = 139.6 Mev, m = 135.0 Mev. Thus the
energy release is 219 Mev.
[Sect. 12.2] RelativisticParticle Kinematics and Dynamics 395
3. Lambda hyperon decays into a neutron or a proton and a pi meson
with a lifetime t = 2.9 x 10 10 sec:
\n + 77°
The rest energy of the lambda hyperon is M = 1115 Mev; that of the
proton m p = 938.5 Mev, and of the neutron m n = 939.8 Mev. With the
pimeson masses given above, we find that the energy release in lambda
decay is 37 Mev in the charged mode and 40 Mev in the neutral mode.
The transformation of a system of mass M at rest into two particles of
mass m 1 and m 2
M^m 1 + m 2 (12.14)
can occur if the initial mass is greater than the sum of the final masses.
We define the mass excess AM:
AM=Mm 1 m 2 (12.15)
The sum of the kinetic energies of the two particles must be equal to AM.
Since the initial system had zero momentum, the two particles must have
equal and opposite momenta, p x = — p 2 = p. From (12.11) the conser
vation of energy can be written
Vp 2 + m 2 + Vp 2 + m 2 = M (12.16)
From this equation it is a straightforward matter to find the magnitude of
the momentum p and the individual particle energies, E 1 and E 2 .
Rather than solve (12.16) we wish to obtain our answers by illustrating
a useful technique which exploits the Lorentz invariance of the scalar
product of two 4 vectors. The conservation of energy and momentum in
the twobody decay can be written as a 4 vector equation :
P=Pi+P2 (12.17)
where the 4vector subscript ju on each symbol has been suppressed. The
squares of the 4vector momenta are the invariants :
(P ■ P) =  M \ ( Pl pj = m 2 , (p 2 p 2 ) = m 2 (12.18)
In (12.18) we have written the squares of the 4vectors as selfscalar
products in order to distinguish the square of a vectorial quantity as a
threespace selfscalar product (e.g., p 2 = p • p). Using (12.17), we form
the square of the 4vector p 2 :
(PzPz) = (PPi)(Pp 1 ) }
(12.19)
or m 2 = M 2  m 2  2(P Pl ) J
M 2 + m 2 
2
m 2 
2M
M 2 + m 2 
m 2
396 Classical Electrodynamics
The scalar product (P • p x ) is Lorentz invariant. In the frame in which the
system M is at rest its space part vanishes, and it has the value :
(Pp 1 ) = ME X (12.20)
Therefore the total energy of the particle with mass m x is
E x = M " + m ± ~ m2 (12.21)
Similarly
E 2 = "' ' "'* '" 1 (12.22)
2M
Often it is more convenient to have expressions for the kinetic energies
than for the total energies. Using (12.15), it is easy to show that
Ti = AM (l^^), i_l f 2 (12.23)
where AM is the mass excess. The term AM/2M is a relativistic correction
absent in the nonrelativistic result. Although it may not have obvious
relativistic origin, a moment's thought shows that, if AM/2M is appreciable
compared to unity, then necessarily the outgoing particles must be treated
relativistically.
As a numerical illustration we consider the first example listed above,
the decay of the pi meson. The mass excess is 33.9 Mev, while M = 139.6
Mev, m^ = 105.7 Mev, m v = 0. Consequently the mumeson and neutrino
kinetic energies are
Tf „ 33.9(l i2LZ £?) = 4.1 Mev
M \ 139.6 2(139.6)/
T v = 33.9  7; = 29.8 Mev
The unique energy of 4. 1 Mev for the mu meson was the characteristic of
pimeson decay at rest which led to its discovery in 1947 by Powell and
coworkers from observations in photographic emulsions.
The lambda particle was first observed in flight by its charged decay
products (p + 77~) in cloud chambers. The charged particle tracks appear
as shown in Fig. 12.1. The particles' initial momenta and identities can
be inferred from their ranges and their curvatures in a magnetic field (or
by other techniques, such as grain counting, in emulsions). The opening
angle 6 between the tracks provides the other datum required to determine
the unseen particle's mass. Consider the square of (12.17):
(P'P) = (p 1 +p 2 )'(p 1 +p 2 ) (12.24)
This becomes
M 2 = m 2  m 2 + 2( Pl ■ p 2 ) (12.25)
[Sect. 12.3] RelativisticParticle Kinematics and Dynamics 397
Fig. 12.1 Decay of lambda particle in flight.
If the scalar product (p x ■ p^) is evaluated in the laboratory frame, we find
M 2 = m? + m 2 2 + 2E X E 2  2p x p 2 cos (12.26)
where p 1 and/? 2 are the magnitudes of the threedimensional momenta.
In a three or more body decay process the particles do not have unique
momenta, but are distributed in energy in some way. These energy spectra
have definite upper end points which can be determined from the kine
matics in ways similar to those used here (see Problem 12.2).
12.3 Center of Momentum Transformation and Reaction Thresholds
A common problem in nuclear or highenergy physics is the collision
of two particles. Particle 1 (the projectile), with mass m x , momentum
p x = p, and energy E lt is incident on particle 2 (the target) of mass m 2 at
rest in the laboratory. The collision may involve elastic scattering,
® + (D^© + ® 02.27)
where the primes mean that the directions of the particles are in general
different. The collision may, on the other hand, be a reaction
® + (D(D + (!) + ••• (12.28)
in which two or more particles are produced, at least one of which is
different from the incident particles. Elastic scattering is always possible,
but reactions may or may not be energetically possible, depending on the
differences in masses of the particles and the incident energy. To determine
the energetics involved and to see the processes in their simplest form
kinematically it is convenient to transform to a coordinate frame K',
where the projectile and the target have equal and oppositely directed
momenta. This frame is called the center of momentum system (sometimes,
loosely, the center of mass system) and is denoted by CM system. The
scattered particles (or reaction products in a twobody reaction) have
equal and opposite momenta making an angle 0' with the initial momenta.
398
Classical Electrodynamics
Figure 12.2 shows the momentum vectors involved in elastic scattering or
a twobody reaction. For elastic scattering, p' = q', but for a reaction
the magnitude of q' must be determined from conservation of total energy
(including rest energies) in the CM system.
To relate the incident energy and momentum in the laboratory to the
CM variables we can either make a direct Lorentz transformation to K',
determining the transformation velocity v CM from the requirement that
p/ = p' = —p 2 ', or we can use the invariance of scalar products. Adopting
the latter procedure, we consider the invariant scalar product
(Pi + P2) • (Pi + P2) = (Pi + P2) • (Pi + p 2 ')
(12.29)
The lefthand side is to be evaluated in the laboratory, where p 2 = 0, and
the righthand side in the CM system, where p/ + p 2 ' = 0. Consequently
we obtain
f  (E 1 + m 2 ) 2 = (£/ + E 2 'f (12.30)
Using E ± 2 = p 2 + rrij 2 , we find that the total energy in the CM system is
E' = E{ + E 2 ' = (mj 2 + m 2 + 2E x m^ (12.31)
The separate energies El and E 2 ' can be found by considering scalar
products like
Pi ' (Pi + P*) = Pi ' (Pi + P2) (12.32)
This gives
Similarly
E 1 ' =
E' =
E' 2 + m/  m 2 2
IE'
E' 2 + m 2 2  m/
IE'
(12.33)
We note the similarity of these expressions to (12.21) and (12.22). The
magnitude of the momentum p' can be obtained from (12.33):
m 2 p
E'
(12.34)
Fig. 12.2 Momentum vectors in the center
of momentum frame for elastic scattering or
a twobody reaction.
[Sect. 12.3] RelativisticParticle Kinematics and Dynamics 399
The Lorentz transformation parameters v CM and y CM can be found by
noting that p 2 ' = /cm^cm = p' and E t = rcM"V This g ives
El + m2 (12.35)
7cm ~
* CM £ 1 + m 2 ' '^ E'
For nonrelativistic motion the kinetic energy in the CM system reduces
T = E'  (m, + m 2 )  ( ™ 2 \\mtf (12.36)
\m x + m 2 /
Similarly the CM velocity and the momentum in the CM system are
v CM = (=»k W = (^W (12.37)
CM Xmj + mj \m x + m 2 /
We see that we can recover the familiar nonrelativistic results from our
completely relativistic expressions. In the extreme of ultrarelativistic
motion (E x > m 1 and m 2 ) the various quantities take on the approximate
limiting values: E'QE^*
CM_ E,
(12.38)
The energy available in the CM system is seen to increase only as the square
root of the incident energy. This means that it is very difficult to obtain
ultrahigh energies in the CM frame when bombarding stationary targets.
The highestenergy accelerators presently existing (at CERN, near Geneva,
Switzerland, and at Brookhaven, N.Y.) produce protons of approximately
30 Bev. If the target is a stationary nucleon, this means about 7 Bev total
CM energy. To have 30 Bev available in the CM frame it would be
necessary to bombard a stationary nucleon with protons of over 470 Bev !
Considerable effort is being put into designs for socalled colliding or
clashing beam accelerators so that no energy is wasted in CM motion.
In a reaction the initial particles of mass m x and m 2 are transformed into
two or more particles with masses /Wj (i = 3,4, . . .). Let AM be the
difference between the sum of masses finally and the sum of masses
initially:* AM = (m 3 + m 4 + • • •)  K + in.) (12.39)
If AM is positive, the reaction will not occur below a certain incident
kinetic energy T th , called the threshold for the reaction. The criterion for
the reaction just to occur is that there be enough energy available in the
* Note that this definition of AM is the negative of the one used in Section 12.2 for
decay processes.
400 Classical Electrodynamics
CM system that the particles can be created with no kinetic energy. This
means that
(E'\ h = m 1 + w 2 + A¥ (12.40)
Using (12.31), it is easy to show that the incident kinetic energy of the
projectile at threshold is
T„, = AM (l + Si + ^) (12.41)
The first two terms in the parentheses are the nonrelativistic terms, while
the last is a relativistic contribution. To illustrate the reactionthreshold
formula we consider the calculation of the threshold energy for photo
production of neutral pi mesons from protons :
y + p^p + ^
Since the photon has no rest mass, the mass difference is AM == m^ =
135.0 Mev, while the target mass is m 2 = m p = 938.5 Mev. Then the
threshold energy is
135.0 ~
T th = 135.0
1 +
2(938.5).
= 135.0(1.072) = 144.7 Mev
As another example consider the production of a protonantiproton pair
in protonproton collisions :
p+p^p+p+p+p
The mass difference is AM= 2m v = 1.877 Bev. From (12.41) we find
r th = 2m p (l + 1 + 1) = 6m p = 5.62 Bev
In this example we find a factorof3 increase over the actual mass
difference, whereas in the photoproduction example the increase was only
7.2 per cent. Other threshold calculations are left to Problem 12.1.
12.4 Transformation of Scattering or Reaction Momenta and Energies
from CM to Laboratory System
In Fig. 12.2 the various CM momenta for a twobody collision are
shown. The initial momenta and energies (p/ = — p 2 ' = p', E x \ E 2 ') have
already been calculated, (12.33) and (12.34). The final CM momenta and
energies (p 3 ' = p 4 ' = q', E 3 ', E^) can be calculated similarly. Since
energy and momentum are conserved, the 4vector momenta satisfy
Pi +P* =Pz +P* (12.42)
[Sect. 12.4] RelativisticParticle Kinematics and Dynamics 401
Then it is easy to show that the energies of the outgoing particles are
£3' =
£4' =
E* + m 3 2  ml
IE'
E' 2 + m 4 2  m?
IE'
(12.43)
where E' is given by (12.31). The obvious symmetry with (12.33) should
be noted. The CM momentum of the outgoing particles is
"' = f It 1  (H^J] i 1  (")T (1144)
An alternative form of this result is
AE
E'
•L Ex + 2?W)~
\ m 9 / 
(12.45)
where A£ x is the incident projectile's energy in the laboratory above the
threshold energy (12.41):
AE, = T X  T th (12.46)
For elastic scattering where m 3 = m x , m 4 = m 2 , (12.45) obviously reduces
to (12.34).
Since the scattering or reaction is actually observed in the laboratory, it
is necessary to transform back from the CM frame to the laboratory.
Figure 12.3 shows the initial momentum p and the final momenta p 3 and p 4
in the laboratory. The CM momenta in Fig. 12.2 have been thrown
forward by the Lorentz transformation. We can express the laboratory
energy E 3 in terms of CM quantities by the Lorentz transformation v CM ,
using (12.35) and (12.4). If 6' is the CM angle of p 3 ' with respect to the
incident direction, we find
E z = y C M(£ 3 ' + ^cm?' cos 6')
(12.47)
mi«
Fig. 12.3 Momentum vectors
in laboratory for a twobody
process.
402 Classical Electrodynamics
Then an explicit expression is
£3 = XPi + m 2 )[ 1 +
m a
7/ 2 J
+
1 1 1 _ / m 3 + m 4
1 
m, —
T^jf j cos 0' (12.48)
where £' is given by (12.31). To obtain E A we merely interchange m 3 and
m 4 and change 0' into n — 6' (cos 0' > —cos 0').
The relation between angles 0' and 3 can be obtained from the expres
sion
tan0, = ^ = l' sind '
Therefore we find
tan 0„ =
Pm 7cm(«' cos0 ' + ^cm £ 3)
E' sin 6'
where
a =
_ vcmEs'
{E x + m 2 )(cos 0' + a)
1 +
(12.49)
(12.50)
m 3 2 — m 4 2
£' 2
£ i + m 2 f"i _ / ma + m^ ir _ fa,  m 4 \ 2 "j'
(12.51)
We note that a is the ratio of the CM velocity to the velocity of particle 3
in the CM system. Just above threshold, a will be large compared to unity.
This means that, as 6' ranges over all values from ► 77 in the CM system,
3 will be confined to some forward cone, < 3 < max . Figure 12.4
shows the general behavior when a > 1 . The laboratory angle 3 is double
valued if a > 1, with particles emitted forwards and backwards in the CM
system appearing at the same laboratory angle. The two types of particles
can be distinguished by their energies. From (12.48) it is evident that the
particles emitted forwards in the CM frame will be of higher energy than
those emitted backwards. For a < 1, it is evident that the denominator in
(12.50) can vanish for some 0' > (tt/2), implying 3 = (tt/2), and is
negative for large 0'. This means that 3 varies over the full range
(0 < 3 < 77) and is a singlevalued function of 0'. Such a curve is shown
in Fig. 12.4.
Although it is not difficult to relate 0' and 3 through (12.50) and so
obtain E z as a function of 3 from (12.48), it is sometimes convenient to
have an explicit expression for this relationship. Using conservation of
energy and momentum in the laboratory,
Px+ P2=P*+ Pi
(12.52)
[Sect. 12.4] RelativisticParticle Kinematics and Dynamics
403
Fig. 12.4 Laboratory angle 3
of particle 3 versus center of
momentum angle 6' for a < 1
and a > 1.
it is a straightforward, although tedious, matter to obtain the result:
( ,t? . 4 it ■ "h 2 + ™2 2 + ™*  m? \
(E 1 + mJlmiEi H ^ J
±p cos o 3
E % =
[(«A + =^
+ m 2 2 — m 3 2 — m 4
)'
m z 2 m 4 2 — p 2 m 3 * sin 2 o 3
J*
(£ x + /n 2 ) 2  p 2 cos 2 3
(12.53)
Only the values of (12.53) greater than m 3 have physical significance. Both
roots are allowed when a > 1 in (12.50), but only one when a < 1, as can
be readily verified. To obtain £ 4 we merely interchange m z and m 4 and
replace 6 3 by 4 .
For elastic scattering with m 3 = m x , m 4 = m 2 , the above relationships
simplify considerably. The scattering angle in the laboratory is given by
(12.50) with
m 1 \m 1
+ m 1
m 2 (E 1 + m 2 )
(12.54)
In the nonrelativistic limit this reduces to the wellknown result a = m x \m 2 .
The energy lost by the incident particle is A£ = T 4 = E± — m 4 . From
(12.48) we can obtain AE in terms of the CM scattering angle:
AE =
m 2 p 2 (l — cos B')
2m 2 E 1 + m x 2 + m 2 2
(12.55)
404
Classical Electrodynamics
An alternative expression for A£in terms of the laboratory angle of recoil
4 can be found from (12.53):
AE = 2m 2 p 2 cos 2 6 4
2m 2 E 1 + m* + m 2 2 + f sin 2 4 U °'
For a headon collision both expressions take on the maximum value
A£ max = ^— H2 57)
The nonrelativistic value of A^^ is
A£m ^(^w (imi,,l2) (12  58)
showing that all the incident kinetic energy can be transferred in a headon
collision if m 1 = m 2 (true relativistically as well).
An important example of energy transfer occurs in collisions between
incident charged particles and atomic electrons. These electrons can be
treated as essentially at rest. If the incident particle is not an electron,
m x > m 2 . Then the maximum energy transfer can be written approxi
mately as
A£ max ~ 2m 2 \^J = 2m 2 y 2 p 2 (12.59)
where y, p are characteristic of the incident particle. Equation (12.59) is
valid, provided the incident energy is not too large :
£i < (— )m 1 (12.60)
\m 2 /
For mu mesons this limit is 20 Bev ; for protons it is nearly 2000 Bev. For
electronelectron collisions {m x = m 2 = m), the maximum energy transfer
is
A£Sx = (y  l)m (12.61)
12.5 Covariant Lorentz Force Equation; Lagrangian and Hamiltonian
for a Relativistic Charged Particle
In Section 12.1 we considered the Lorentz force equation as a method
of establishing the Lorentz transformation properties of the momentum
and energy of a particle, but we did not explicitly examine the equation as
a covariant equation of motion for a particle moving in external fields. We
[Sect. 12.5] RelativisticParticle Kinematics and Dynamics 405
now want to establish that covariance and discuss the associated Lagran
gian, canonical momenta, and Hamiltonian. From equations (12.2) and
and (11.129) we see that we can write the force equation in the form:
dt c J
F UV J V d 3 x (12.62)
where the volume integral is over the extent of the charge. If the particle's
velocity is v and its total charge e, then
^ = F MV t; v (12.63)
dt c
where v v = v k for v = k = 1, 2, 3 and u 4 = ic. This is not yet a co variant
form for the equation, since v v is not a 4vector, and dpjdt is not one
either. This deficiency can be remedied by writing a derivative with
respect to proper time t (11.62) rather than t. Since dt = y dr, we obtain
^ = iWt; v (12.64)
dr c
But now yv v = pjm is a 4vector (sometimes called the 4velocity). Con
sequently we arrive at the obviously co variant force equation for a particle :
^ = FW> v (12.65)
dr mc
This is the counterpart for a discrete particle of the Lorentz forcedensity
equation (11.129) for continuously distributed charge and current.
Having established its covariance, it is often simplest to revert to the
spacetime forms :
dt \ c
dE ,,
— = e\ • E
dt
(12.66)
in any convenient reference frame. Equation (12.65) shows that, as long
as all the different quantities are transformed according to their separate
transformation laws, the noncovariant forms will be valid in any Lorentz
frame.
Although the force equation (12.65) or (12.66) is sufficient to describe
the general motion of a charged particle in external electromagnetic fields,
it is sometimes convenient to use the ideas and formalism of Lagrangian
and Hamiltonian mechanics. In order to see how to obtain an appropriate
Lagrangian for the Lorentz force equation, we start with a free, but
406 Classical Electrodynamics
relativistic, particle. Since the Lagrangian must be a function of velocities
and coordinates, we write the freeparticle equation of motion as
— (ymy) = (12.67)
where y = [1  (v 2 /c 2 )]^. At the least sophisticated level we know that
the Lagrangian L must be chosen strategically so that the EulerLagrange
equations of motion,
d (dL\ dL „
ifa)*, (12  68)
are the same as Newton's equations of motion. Only a moment's con
sideration shows that a suitable Lagrangian for a free particle is
L f =mc 2 \lJ (12.69)
Evidently this form yields (12.67) when substituted into (12.68).
To obtain the freeparticle Lagrangian in a more elegant way we
consider Hamilton's principle or the principle of least action. This
principle states that the motion of a mechanical system is such that in
going from one configuration a at time t x to another configuration b at
time t 2 , the action integral A, defined as the time integral of the Lagrangian
along the path of the system,
J*'
A = \ Ldt (12.70)
J a
is an extremum (actually a minimum). By considering small variations of
the path taken and demanding 6A = 0, one obtains the EulerLagrange
equations of motion (12.68). We now appeal to the Lorentz invariance of
the action in order to determine the freeparticle Lagrangian. That the
action is a Lorentz scalar follows the first postulate of relativity, since the
requirement that it be an extremum determines the mechanical equations
of motion. If we introduce the proper time through dt = y dr, the action
integral becomes :
(yL) dr (12.71)
Since proper time is Lorentz invariant, the condition that A be also
Lorentz invariant forces yL to be Lorentz invariant. This is a general
condition on the Lagrangian. For a free particle L f can be a function of
only the velocity of the particle (and perhaps its mass). The only Lorentz
invariant quantities involving the velocity are functions of the 4vector
scalar product (l/m 2 )(p • p), where p^ is the 4momentum of the particle.
[Sect. 12.5] RelativisticParticle Kinematics and Dynamics 407
Since (p ■ p) = —m 2 , we see that for a free particle yL f is a constant,
yL f = I (12.72)
Then the action is proportional to the integral of the proper time over the
path from the initial spacetime point a to the final spacetime point b.
This integral is Lorentz invariant, but depends on the path taken. For
purposes of calculation, consider a reference frame in which the particle
is initially at rest. From definition (11.62) of proper time it is clear that,
if the particle stays at rest in that frame, the integral over proper time will
be larger than if it moves with a nonzero velocity along its path. Con
sequently we see that a straight world line joining the initial and final
points of the path gives the maximum integral over proper time, or, with
the negative sign in (12.72), a minimum for the action integral. Com
parison with Newton's equation for nonrelativistic motion shows that
X = mc 2 , yielding the freeparticle Lagrangian (12.69).
The general requirement that yL be Lorentz invariant allows us to
determine the Lagrangian for a relativistic charged particle in external
electromagnetic fields, provided we know something about the Lagrangian
(or equations of motion) for nonrelativistic motion in static fields. A
slowly moving charged particle is influenced predominantly by the electric
field which is derivable from the scalar potential O. The potential energy
of interaction is V = eO. Since the nonrelativistic Lagrangian is (T — V),
the interaction part L iQt of the relativistic Lagrangian must reduce in the
nonrelativistic limit to
Lmt — L?S = e<D (12.73)
Our problem thus becomes that of finding a Lorentz invariant expression
for yL int which reduces to (12.73) for nonrelativistic velocities. Since O
is the fourth component of the 4vector potential A M , we anticipate that
yL int will involve the scalar product of A^ with some 4vector. The only
other 4vectors available are the momentum and position vectors of the
particle. Since gamma times the Lagrangian must be translationally
invariant as well as Lorentz invariant, it cannot involve the coordinates
explicitly. Hence the interaction Lagrangian must be*
Lmt =  — (p ■ A) = ^^  e<D (12.74)
y mc c
* Without appealing to the nonrelativistic limit this form of L lnt can be written down
by demanding that yL lDt be a Lorentz invariant which is (1) linear in the charge of the
particle, (2) linear in the electromagnetic potentials, (3) translationally invariant, and
(4) a function of no higher than the first time derivative of the particle coordinates. The
reader may consider the possiblity of an interaction Lagrangian satisfying these condi
tions, but linear in the field strengths F^, rather than the potentials A^.
408 Classical Electrodynamics
where the coefficient of the scalar product (p • A) is chosen to yield (12.73)
in the limit v — ► 0.
The combination of (12.69) and (12.74) yields the complete relativistic
Lagrangian for a charged particle :
ri
l= \
— mc 2 H (p A)
yL mc
(12.75)
mc 2 i _l + vAeO
where the upper (lower) line gives L in 4vector (explicit spacetime) form.
Verification that (12.75) does indeed lead to the Lorentz force equation
will be left as an exercise for the reader. Use must be made of the con
vective derivative [d/dt = (d/dt) + v • V] and the standard definitions of
the fields in terms of the potentials.
The canonical momentum P conjugate to the position coordinate x is
obtained by the definition,
P i = — = ymv i + A i (12.76)
dv i c
Thus the conjugate momentum is
P = p + A (12.77)
c
where p = ym\ is the momentum in the absence of fields. The Hamiltonian
H is a function of the coordinate x and its conjugate momentum P and is a
constant of the motion if the Lagrangian is not an explicit function of
time. The Hamiltonian is defined in terms of the Lagrangian as
H = P • v  L (12.78)
The velocity v must be eliminated from (12.78) in favor of P and x. From
(12.76) or (12.77) we find that
cP  eA
v =
f)
^ 22 (1279)
When this is substituted into (12.78) and into L (12.75), the Hamiltonian
takes on the form :
H = V(cP  eA) 2 + w 2 c 4 + e<S> (12.80)
Again the reader may verify that Hamilton's equations of motion can be
combined to yield the Lorentz force equation. Equation (12.80) is an
[Sect. 12.6] RelativisticParticle Kinematics and Dynamics 409
expression for the total energy W of the particle. It differs from the free
particle energy by the addition of the potential energy e® and by the
replacement p > [P — 0/c)A]. These two modifications are actually only
one 4 vector change. This can be seen by transposing eO in (12.80) and
squaring both sides. Then
(cP  eXf {W e<S>f = {mc*f (12.81)
This is just the 4vector scalar product
(p.p)=(mcf (12.82)
where ft _(^).[(,_2).v_
eO)
(12.83)
We see that in some sense the total energy Wis the fourth component of a
canonically conjugate 4momentum of which (12.77) is the space part. An
alternative formulation with a relativistically invariant Lagrangian which
is a function of the 4 velocity u M = pjm is discussed in Problem 12.5. There
the canonical 4momentum arises naturally.
The Lagrangian and Hamiltonian formulation of the dynamics of a
charged particle has been outlined for several reasons. One is that the
concept of Lorentz invariance, coupled with other physical requirements,
was shown to be a powerful tool in the systematic construction of a
Lagrangian which yields dynamic equations of motion. Another is that
the Lagrangian is often a convenient starting point in discussing particle
dynamics. Finally, the concepts and ideas of conjugate variables, etc.,
are useful even when one proceeds to solve the force equation directly.
12.6 LowestOrder Relativistic Corrections to the Lagrangian for
Interacting Charged Particles
In the previous section we discussed the general Lagrangian formalism
for a relativistic particle in external electromagnetic fields described by the
vector and scalar potentials, A and O. The appropriate interaction
Lagrangian was given by (12.74). If we now consider the problem of a
Lagrangian description of the interaction of two or more charged particles
with each other, we see that it is possible only at nonrelativistic velocities.
The Lagrangian is supposed to be a function of the instantaneous velocities
and coordinates of all the particles. When the finite velocity of propa
gation of electromagnetic fields is taken into account, this is no longer
possible, since the values of the potentials at one particle due to the other
particles depend on their state of motion at "retarded" times. Only when
•410 Classical Electrodynamics
retardation effects can be neglected is a Lagrangian description of the
system of particles alone possible. In view of this one might think that a
Lagrangian could be formulated only in the static limit, i.e., to zeroth
order in (v/c). We will now show, however, that lowestorder relativistic
corrections can be included, giving an approximate Lagrangian for inter
acting particles, correct to the order of (v/c) 2 inclusive.
It is sufficient to consider two interacting particles with charges q x and
q 2 , masses m 1 and m 2 , and coordinates x x and x 2 . The relative separation
is r = x x — x 2 . The interaction Lagrangian in the static limit is just the
negative of the electrostatic potential energy,
L NR = _ M2 (12g4)
r
If attention is directed to the first particle, this can be viewed as the negative
of the product of q x and the scalar potential 12 due to the second particle
at the position of the first. This is of the same form as (12.73). If we wish
to generalize beyond the static limit, we must, according to (12.74),
determine both 12 and A 12 , at least to some degree of approximation. In
general there will be relativistic corrections to both <I> 12 and A 12 . But in
the Coulomb gauge, the scalar potential is given correctly to all orders in
v/c by the instantaneous Coulomb potential. Thus, if we calculate in that
gauge, the scalarpotential contribution 12 is already known. All that
needs to be considered is the vector potential A 12 .
If only the lowestorder relativistic corrections are desired, retardation
effects can be neglected in computing A 12 . The reason is that the vector
potential enters the Lagrangian (12.74) in the combination q x (y x lc) • A 12 .
Since A 12 itself is of the order of vjc, greater accuracy in calculating A 12
is unnecessary. Consequently, we have the magnetostatic expression,
^luwdv (1285)
c J \x x — x'l
where J t is the transverse part of the current due to the second particle,
as discussed in Section 6.5. From equations (6.46)(6.50) it is easy to see
that the transverse current is
J t (x') = 4 2 v 2 *x'  x 2 )  q f V ( V2 : (X '~f ) (12.86)
Att \ x — X«5 /
When this is inserted in (12.85), the first term can be integrated immediately.
Thus
A ~ 42y_ 2 _ qz_ f 1 v , / v 2 (x'x 2 ) \ dZx ,
12 cr Attc J x'  Xl  \ x'  x 2  3 /
[Sect. 12.7] RelativisticParticle Kinematics and Dynamics 411
By changing variables to y = x'  x 2 and integrating by parts, the integral
can be put in the form,
A qft _ 31. V r f ^ i— d*y (12.88)
cr 4ttc J f y  r
The integral can now be done in a straightforward manner to yield
A ~?2
■i\.ia
2 v r
Lr
(¥)1
The differentiation of the second term leads to the final result
r(y 2 • r)
g2
2cr
v 2 +
(12.89)
(12.90)
With expression (12.90) for the vector potential due to the second
particle at the position of the first, the interaction Lagrangian for two
charged particles, including lowestorder relativistic effects, is
r _ 4i<7 2 ( 1 , J_
v . v , (vir)(v 2 .ry
1 r 2
(12.91)
This interaction form was first obtained by Darwin in 1920. It is of
importance in a quantummechanical discussion of relativistic corrections
in twoelectron atoms. In the quantummechanical problem the velocity
vectors are replaced by their corresponding quantummechanical operators
(Dirac a's). Then the interaction is known as the Breit interaction (1930).
12.7 Motion in a Uniform, Static, Magnetic Field
As a first important example of the dynamics of charged particles in
electromagnetic fields we consider the motion in a uniform, static,
magnetic induction B. The equations of motion (12.66) are
dp e _ dE _
K = vxB, — =
dt
dt
(12.92)
Since the energy is constant in time, the magnitude of the velocity is
constant and so is y. Then the first equation can be written
d\
dt
= V X CO]
where
t»B =
eB _ ecB
ymc E
(12.93)
(12.94)
412 Classical Electrodynamics
is the gyration or precession frequency. The motion described by (12.93)
is a circular motion perpendicular to B and a uniform translation parallel
to B. The solution for the velocity is easily shown to be
v(t) = v u e 3 + co B a(e 1  it^e'™* 1 (12.95)
where c 3 is a unit vector parallel to the field, e x and e 2 are the other
orthogonal unit vectors, y„ is the velocity component along the field, and
a is the gyration radius. The convention is that the real part of the equation
is to be taken. Then one can see that (12.95) represents a counterclockwise
rotation (for positive charge e) when viewed in the direction of B. Another
integration yields the displacement of the particle,
x(r) = Xo + v H te 3 + iafa  i^)e iu>Bt (12.96)
The path is a helix of radius a and pitch angle a = tan 1 (vJa> B a). The
magnitude of the gyration radius a depends on the magnetic induction B
and the transverse momentum p x of the particle. From (12.94) and (12.95)
it is evident that
cp L = eBa
This form is convenient for the determination of particle momenta. The
radius of curvature of the path of a charged particle in a known B allows
the determination of its momentum. For particles with charge the same
in magnitude as the electronic charge, the momentum can be written
numerically as
p ± (Mev/c) = 3.00 x lO" 4 ^ (gausscm) (12.97)
12.8 Motion in Combined, Uniform, Static Electric and Magnetic Fields
We now consider a charged particle moving in a combination of electric
and magnetic fields E and B, both uniform and static, but in general not
parallel. As an important special case, perpendicular fields will be treated
first. The force equation (12.66) shows that the particle's energy is not
constant in time. Consequently we cannot obtain a simple equation for
the velocity, as was done for a static magnetic field. But an appropriate
Lorentz transformation simplifies the equations of motion. Consider a
Lorentz transformation to a coordinate frame K' moving with a velocity
u with respect to the original frame. Then the Lorentz force equation for
the particle in K' is
d^
dt'
t' \ C 1
[Sect. 12.8] RelativisticParticle Kinematics and Dynamics 413
where the primed variables are referred to the system K'. The fields E'
and B' are given by relations (11.115) with v replaced by u, where  and i
refer to the direction of u. Let us first suppose that E < B. If u is
now chosen perpendicular to the orthogonal vectors E and B,
u = c
(E x B)
B 2
(12.98)
we find the fields in K' to be
(12.99)
E„' = 0, E x ' = ylE +  xBl =0
b,'o, b.'IbP^J'b
In the frame K' the only field acting is a static magnetic field B' which
points in the same direction as B, but is weaker than B by a factor y~ x .
Thus the motion in K' is the same as that considered in the previous
section, namely a spiraling around the lines of force. As viewed from the
original coordinate system, this gyration is accompanied by a uniform
"drift" u perpendicular to E and B given by (12.98). This drift is sometimes
called the E x B drift. It has already been considered for a conducting
fluid in another context in Section 10.3. The drift can be understood
qualitatively by noting that a particle which starts gyrating around B is
accelerated by the electric field, gains energy, and so moves in a path with
a larger radius for roughly half of its cycle. On the other half, the electric
field decelerates it, causing it to lose energy and so move in a tighter arc.
The combination of arcs produces a translation perpendicular to E and B
as shown in Fig. 12.5. The direction of drift is independent of the sign of
the charge of the particle.
The drift velocity u (12.98) has physical meaning only if it is less than
the velocity of light, i.e., only if E < B. If E > B, the electric field
Fig. 12.5 E x B drift of charged
particles in crossed fields.
414 Classical Electrodynamics
is so strong that the particle is continually accelerated in the direction of E
and its average energy continues to increase with time. To see this we
consider a Lorentz transformation from the original frame to a system K"
moving with a velocity
(Ex B)
£ 2
u' = c ^^ (12.100)
relative to the first. In this frame the electric and magnetic fields are
1 /f 2 — R 2 Y^
v = o, e/ = e=(^)e
B„" = 0, B x " = y'\B  Hl^Zj = o
(12.101)
Thus in the system K" the particle is acted on by a purely electrostatic
field which causes hyperbolic motion with everincreasing velocity (see
Problem 12.7).
The fact that a particle can move through crossed E and B fields with
the uniform velocity u = cE/B provides the possibility of selecting charged
particles according to velocity. If a beam of particles having a spread in
velocities is normally incident on a region containing uniform crossed
electric and magnetic fields, only those particles with velocities equal to
cEjB will travel without deflection. Suitable entrance and exit slits will
then allow only a very narrow band of velocities around cE/B to be
transmitted, the resolution depending on the geometry, the velocities
desired, and the field strengths. When combined with momentum
selectors, such as a deflecting magnet, these E x B velocity selectors can
separate a very pure and monoenergetic beam of particles of a definite
mass from a mixed beam of particles with different masses and momenta.
Largescale devices of this sort are commonly used to provide experimental
beams of particles produced in very highenergy accelerators.
If E has a component parallel to B, the behavior of the particle cannot
be understood in such simple terms as above. The scalar product E • B is a
Lorentz invariant quantity (see Problem 1 1.10), as is (2? 2 — E 2 ). When the
fields were perpendicular (E • B = 0), it was possible to find a Lorentz
frame where E = Oif B > E,orB = Oif E > B. In those coordinate
frames the motion was relatively simple. If E B^0, electric and magnetic
fields will exist simultaneously in all Lorentz frames, the angle between the
fields remaining acute or obtuse depending on its value in the original
coordinate frame. Consequently motion in combined fields must be
considered. When the fields are static and uniform, it is a straightforward
matter to obtain a solution for the motion in cartesian components. This
will be left for Problem 12.10.
[Sect. 12.9] RelativisticParticle Kinematics and Dynamics
415
12.9 Particle Drifts in Nonuniform, Static Magnetic Fields
In astrophysical and thermonuclear applications it is of considerable
interest to know how particles behave in magnetic fields which vary in
space. Often the variations are gentle enough that a perturbation solution
to the motion, first given by Alfven, is an adequate approximation.
"Gentle enough" generally means that the distance over which B changes
appreciably in magnitude or direction is large compared to the gyration
radius a of the particle. Then the lowestorder approximation to the
motion is a spiraling around the lines of force at a frequency given by the
local value of the magnetic induction. In the next approximation, slow
changes occur in the orbit which can be described as a drifting of the
guiding center.
The first type of spatial variation of the field to be considered is a
gradient perpendicular to the direction of B. Let the gradient at the point
of interest be in the direction of the unit vector n, with n • B = 0. Then,
to first order, the gyration frequency can be written
Wb(x) =
ymc
B(x)
tOr
1 +
Bo Wo
(12.102)
In (12.102) £ is the coordinate in the direction n, and the expansion is
about the origin of coordinates where a> B = co . Since the direction of Bis
unchanged, the motion parallel to B remains a uniform translation.
Consequently we consider only modifications in the transverse motion.
Writing v ± = v + v 1? where v is the uniformfield transverse velocity and
v x is a small correction term, we can substitute (12.102) into the force
equation
— ± = Vj. X U) B (X)
dt
(12.103)
and, keeping only firstorder terms, obtain the approximate result,
d\ 1
dt
_l / ^ ( dB \
v 1 + v (n.x )— — I
B n \ dt /,
B \df /„.
X tO n
(12.104)
From (12.95) and (12.96) it is easy to see that for a uniform field the
transverse velocity v and coordinate x are related by
v = w x (x  X)
(Xq  X) = — (co x v )
(12.105)
416 Classical Electrodynamics
where X is the center of gyration of the unperturbed circular motion
(X = here). If (o> x v ) is eliminated in (12.104) in favor of x , we
obtain
dt
1 ~ y 1  — I — I c*) x x (n • x ) x u> (12.106)
\ L B \d£/ 1
This shows that, apart from oscillatory terms, \ x has a non zero average
value,
y G = <v x > = i (?f ) to x ((XoUn . x )> (12.107)
To determine the average value of (x )j(n • x ), it is necessary only to
observe that the rectangular components of (x ) ± oscillate sinusoidally
with peak amplitude a and a phase difference of 90°. Hence only the
component of (x ) ± parallel to n contributes to the average, and
<(x )i(n • x )> =  n
Thus the gradient drift velocity is given by
a 2 1 dB ,
(12.108)
(12.109)
An alternative form, independent of coordinates, is
(12.110)
From (12.110) it is evident that, if the gradient of the field is such that a
oi B a 2d
+e
VB
Fig. 12.6 Drift of charged par
ticles due to transverse gradient
of magnetic field.
[Sect. 12.9] RelativisticParticle Kinematics and Dynamics
y
417
y
Bo
* " ^
n£KKK ( <CKKKK <
bUuu(
JUUUU x
(a)
(b)
Fig. 12.7 (a) Particle moving in helical path along lines of uniform, constant magnetic
induction, (b) Curvature of lines of magnetic induction will cause drift perpendicular
to the (x, y) plane.
\VB/B\ < 1, the drift velocity is small compared to the orbital velocity
(w B a). The particle spirals rapidly while its center of rotation moves slowly
perpendicular to both B and VB. The sense of the drift for positive
particles is given by (12.110). For negatively charged particles the sign of
the drift velocity is opposite ; the sign change comes from the definition
of co B . The gradient drift can be understood qualitatively from considera
tion of the variation of gyration radius as the particle moves in and out of
regions of larger than average and smaller than average field strength.
Figure 12.6 shows this qualitative behavior for both signs of charge.
Another type of field variation which causes a drifting of the particle's
guiding center is curvature of the lines of force. Consider the two
dimensional field shown in Fig. 12.7. It is locally independent of z. On
the lefthand side of the figure is a constant, uniform magnetic induction
B , parallel to the x axis. A particle spirals around the field lines with a
gyration radius a and a velocity co B a, while moving with a uniform velocity
v u along the lines of force. We wish to treat that motion as a zeroorder
approximation to the motion of the particle in the field shown on the right
hand side of the figure, where the lines of force are curved with a local
radius of curvature R which is large compared to a.
The firstorder motion can be understood as follows. The particle tends
to spiral around a field line, but the field line curves off to the side. As far
as the motion of the guiding center is concerned, this is equivalent to a
centrifugal acceleration of magnitude v^jR. This acceleration can be
viewed as arising from an effective electric field,
_ ym R 2
e R 2
(12.111)
418 Classical Electrodynamics
in addition to the magnetic induction B . From (12.98) we see that the
combined effective electric field and the magnetic induction cause a
curvature drift velocity,
With the definition of co B = eB^ymc, the curvature drift can be written
y r = llL(*lL*!>) (12.113)
co B R\ RB J
The direction of drift is specified by the vector product, in which R is the
radius vector from the effective center of curvature to the position of the
charge. The sign in (12.113) is appropriate for positive charges and is
independent of the sign of y„. For negative particles the opposite sign
arises from co B .
A more straightforward, although pedestrian, derivation of (12.113) can
be given by solving the Lorentz force equation directly. If we use cylin
drical coordinates (p, cf>, z) appropriate to Fig. 1 2.1b with origin at the center
of curvature, the magnetic induction has only a <f> component, B^ = B .
Then the force equation can be easily shown to give the three equations,
p — p(f> 2 = — co B z
p$ + 2pcf> = (12.114)
z = a> B p
If the zeroorder trajectory is a helix with radius a small compared to the
radius of curvature R, then, to lowest order, (j> ~ vJR, while p ~ R. Thus
the first equation of (12.114) yields an approximate result for z:
• ■ V " (12.115)
(OnR
This is just the curvature drift given by (12.113).
For regions of space in which there are no currents the gradient drift
y (12.110) and the curvature drift v c (12.113) can be combined into one
simple form. This follows from the fact that V x B = implies
?±2 =   (12.116)
BR 2
Evidently then the sum of v G and \ c is a general drift velocity,
Vz, =  J ^(^, 2 + K 2 )( R ^) (12.117)
(o n R \ RB I
[Sect. 12.10] RelativisticP article Kinematics and Dynamics 419
where y = co B a is the transverse velocity of gyration. For singly charged
nonrelativistic particles in thermal equilibrium, the magnitude of the drift
velocity is
^(cm/sec) = p , 17 ^ ( ° K) , (12118)
R(m) B(gauss)
The particle drifts implied by (12.117) are troublesome in certain types
of thermonuclear machines designed to contain hot plasma. A possible
configuration is a toroidal tube with a strong axial field supplied by
solenoidal windings around the torus. With typical parameters of R = 1
meter, B = 10 3 gauss, particles in a 1ev plasma (T~ lO^K) will have
drift velocities v D ^ > 1.8 x 10 3 cm/sec. This means that they will drift out
to the walls in a small fraction of a second. For hotter plasmas the drift
rate is correspondingly greater. One way to prevent this firstorder drift
in toroidal geometries is to twist the torus into a figure eight. Since the
particles generally make many circuits around the closed path before
drifting across the tube, they feel no net curvature or gradient of the field.
Consequently they experience no net drift, at least to first order in \\R.
This method of eliminating drifts due to spatial variations of the magnetic
field is used in the Stellarator type of thermonuclear machine, in which
containment is attempted with a strong, externally produced, axial
magnetic field, rather than a pinch (see Sections 10.510.7).
12.10 Adiabatic Invariance of Flux through Orbit of Particle
The various motions discussed in the previous sections have been
perpendicular to the lines of magnetic force. These motions, caused by
electric fields or by the gradient or curvature of the magnetic field, arise
because of. the peculiarities of the magneticforce term in the Lorentz force
equation. To complete our general survey of particle motion in magnetic
fields we must consider motion parallel to the lines of force. It turns out
that for slowly varying fields a powerful tool is the concept of adiabatic
invariants. In celestial mechanics and in the old quantum theory adiabatic
invariants were useful in discussing perturbations on the one hand, and in
deciding what quantities were to be quantized on the other. Our discussion
will resemble most closely the celestial mechanical problem, since we are
interested in the behavior of a charged particle in slowly varying fields
which can be viewed as small departures from the simple, uniform, static
field considered in Section 12.7.
The concept of adiabatic invariance is introduced by considering the
action integrals of a mechanical system. If q t and p+ are the generalized
420 Classical Electrodynamics
canonical coordinates and momenta, then, for each coordinate which is
periodic, the action integral / t is defined by
Ji^fPidq, (12.119)
The integration is over a complete cycle of the coordinate q t . For a given
mechanical system with specified initial conditions the action integrals J {
are constants. If now the properties of the system are changed in some way
(e.g., a change in spring constant or mass of some particle), the question
arises as to how the action integrals change. It can be proved* that, if
the change in property is slow compared to the relevant periods of motion
and is not related to the periods (such a change is called an adiabatic
change), the action integrals are invariant. This means that, if we have a
certain mechanical system in some state of motion and we make an
adiabatic change in some property so that after a long time we end up with
a different mechanical system, the final motion of that different system will
be such that the action integrals have the same values as in the initial
system. Clearly this provides a powerful tool in examining the effects of
small changes in properties.
For a charged particle in a uniform, static, magnetic induction B the
transverse motion is periodic. The action integral for this transverse
motion is
J = <t>P ± .<H, (12.120)
where P is the transverse component of the canonical momentum (12.77)
and d\ is a directed line element along the circular path of the particle.
From (12.77) we find that
J = &) ym\ ± • dl +  <b Adl (12.121)
Since v ± is parallel to dl, we find
J = d> ymco B a 2 dd +  <P A . d\ (12.122)
Applying Stokes's theorem to the second integral and integrating over 6
in the first integral, we obtain
J = 27rymco B a 2 +  B • n da (12.123)
c Js
* See, for example, M. Born, The Mechanics of the Atom, Bell, London (1927).
[Sect. 12.10] RelativisticPartick Kinematics and Dynamics
421
Since the line element d\ in (12.120) is in a counterclockwise sense relative
to B, the unit vector n is antiparallel to B. Hence the integral over the
circular orbit subtracts from the first term. This gives
J = ymco B 7ra 2 =  (Bira 2 )
(12.124)
making use of a> B = eB/ymc. The quantity Bna 2 is the flux through the
particle's orbit.
If the particle moves through regions where the magnetic field strength
varies slowly in space or time, the adiabatic invariance of /means that the
flux linked by the particle's orbit remains constant. If B increases, the
radius a will decrease so that Bna 2 remains unchanged. This constancy of
flux linked can be phrased in several ways involving the particle's orbit
radius, its transverse momentum, its magnetic moment. These different
statements take the forms :
Ba 2
yp
► are adiabatic invariants
(12.125)
where \x = {ew B a 2 j2c) is the magnetic moment of the current loop of the
particle in orbit. If there are only static magnetic fields present, the speed
of the particle is constant and its total energy does not change. Then the
magnetic moment /u is itself an adiabatic invariant. In timevarying fields
or with static electric fields, /u is an adiabatic invariant only in the
nonrelativistic limit.
Let us now consider a simple situation in which a static magnetic field
B acts mainly in the z direction, but has a small positive gradient in that
direction. Figure 12.8 shows the general behavior of the lines of force. In
addition to the z component of field there is a small radial component due
to the curvature of the lines of force. For simplicity we assume cylindrical
symmetry. Suppose that a particle is spiraling around the z axis in an
Fig. 12.8
422 Classical Electrodynamics
orbit of small radius with a transverse velocity \ xo and a component of
velocity y 0 parallel to B at z = 0, where the axial field strength is B . The
speed of the particle is constant so that at any position along the z axis
v n 2 + v 1 2 = v ( ? (12.126)
where v Q 2 = v 10 2 + u 0 2 is the square of the speed at z = 0. If we assume
that the flux linked is a constant of the motion, then (12.125) allows us to
write „ 2
(12.127)
o
B B t
where B is the axial magnetic induction. Then we find the parallel velocity
at any position along the z axis given by
V = V»Lo 2 ^r (12128)
Equation (12.128) for the velocity of the particle in the z direction is
equivalent to the first integral of Newton's equation of motion for a
particle in a onedimensional potential*
V(z) = \m V ^ B(z)
B o
If B(z) increases enough, eventually the righthand side of (12.128) will
vanish at some point z = z . This means that the particle spirals in an
evertighter orbit along the lines of force, converting more and more
translational energy into energy of rotation, until its axial velocity vanishes.
Then it turns around, still spiraling in the same sense, and moves back in
the negative z direction. The particle is reflected by the magnetic field, as
is shown schematically in Fig. 12.9.
Equation (12.128) is a consequence of the assumption that p 2 \B is
invariant. To show that at least to first order this invariance follows
directly from the Lorentz force equation, we consider an explicit solution
of the equations of motion. If the magnetic induction along the axis is
B(z), there will be a radial component of the field near the axis given by
the divergence equation as
dB(z)
dz
B p ( P ,z)~y u ^ (12.129)
where p is the radius out from the axis. The z component of the force
equation is
(pj>B.)~^ ^i>"=P (12.130)
ymc Lyme dz
* Note, however, that our discussion is fully relativistic. The analogy with one
dimensional nonrelativistic mechanics is only a formal one.
[Sect. 12.10] RelativisticParticle Kinematics and Dynamics
423
Fig. 12.9 Reflection of charged
particle out of region of high field
strength.
where <j> is the angular velocity around the z axis. This can be written,
correct to first order in the small variation of B(z), as
z~ —
Vw 2 dB(z)
2B n dz
(12.131)
where we have used p 2 ~ — (a 2 o) B ) = — {v ± qJo) b ^. Equation (12.131) has
as its first integral equation (12.128), showing that to first order in small
quantities the constancy of flux Unking the orbit follows directly from the
equations of motion.
The adiabatic invariance of the flux linking an orbit is useful in discussing
particle motions in all types of spatially varying magnetic fields. The
simple example described above illustrates the principle of the "magnetic
mirror" : charged particles are reflected by regions of strong magnetic
field. This mirror property formed the basis of a theory of Fermi for the
acceleration of cosmicray particles to very high energies in interstellar
space by collisions with moving magnetic clouds. The mirror principle
can be applied to the containment of a hot plasma for thermonuclear
energy production. A magnetic bottle can be constructed with an axial
field produced by solenoidal windings over some region of space, and
additional coils at each end to provide a much higher field towards the
ends. The lines of force might appear as shown in Fig. 12.10. Particles
created or injected into the field in the central region will spiral along the
axis, but will be reflected by the magnetic mirrors at each end. If the
ratio of maximum field B m in the mirror to the field B in the central region
is very large, only particles with a very large component of velocity parallel
to the axis can penetrate through the ends. From (12.128) is it evident that
the criterion for trapping is
M < (*=  if
v xo I \B /
(12.132)
424
Classical Electrodynamics
Fig. 12.10 Schematic diagram
of "mirror" machine for the
containment of a hot plasma.
If the particles are injected into the apparatus, it is easy to satisfy require
ment (12.132). Then the escape of particles is governed by the rate at
which they are scattered by residual gas atoms, etc., in such a way that
their velocity components violate (12.132).
Another area of application of these principles is to terrestrial and
stellar magnetic fields. The motion of charged particles in the magnetic
dipole fields of the sun or earth can be understood in terms of the adiabatic
invariant discussed here and the drift velocities of Section 12.9. Some
aspects of this topic are left to Problems 12.11 and 12.12 on the trapped
particles around the earth (the Van Allen belts).
REFERENCES AND SUGGESTED READING
The applications of relativistic kinematics, apart from precision work in lowenergy
nuclear physics, all occur in the field of highenergy physics. In books on that field, the
relativity is taken for granted, and calculations of kinematics are generally omitted or
put in appendices. One exception is the book by
Baldin, Gol'danskii, and Rozenthal
which covers the subject exhaustively with many graphs.
The Lagrangian and Hamiltonian formalism for relativistic charged particles is treated
in every advanced mechanics textbook, as well as in books on electrodynamics. Some
useful references are
Corben and Stehle, Chapter 16,
Goldstein, Chapter 6,
Landau and Lifshitz, Classical Theory of Fields, Chapters 2 and 3.
The motion of charged particles in external electromagnetic fields, especially inhomo
geneous magnetic fields, is an increasingly important topic in geophysics, solar physics,
and thermonuclear research. The classic reference for these problems is
Alfven,
but the basic results are also presented by
Chandrasekhar, Chapters 2 and 3,
Linhart, Chapter 1,
Spitzer, Chapter 1.
[Probs. 12] RelativisticParticle Kinematics and Dynamics 425
Another important application of relativistic chargedparticle dynamics is to high
energy accelerators. An introduction to the physics problems of this field will be found
in
Corben and Stehle, Chapter 17,
Livingston.
For a more complete and technical discussion, with references, consult
E. D. Courant and H. S. Snyder, Ann. Phys., 3, 1 (1958).
PROBLEMS
12.1 Use the transformation to center of momentum coordinates to determine
the threshold kinetic energies in Mev for the following processes :
(a) pimeson production in nucleonnucleon collisions (m„/M = 0.15),
(b) pimeson production in pi mesonnucleon collisions,
(c) pimeson pair production in nucleonnucleon collisions,
id) nucleonpair production in electronelectron collisions.
12.2 If a system of mass M decays or transforms at rest into a number of
particles, the sum of whose masses is less than M by an amount AM,
(a) show that the maximum kinetic energy of the /th particle (mass m f ) is
(7VU,u«.(i3£)
(b) determine the maximum kinetic energies in Mev and also the ratios
to AMc 2 of each of the particles in the following decays or transformations
of particles at rest:
p »■ e + v + v
KT —*■ TfT~ + 1T~ + 77+
K± >e± + 77° + v
K±
A«±
+ 7T
+ v
p
+ P
*2tt+
+ 2tt
+7T
p
+ P
— K+ + K~
+ 3tt°
12.3 A pi meson (m^ 2 = 140 Mev) collides with a proton (m 2 c 2 = 938 Mev)
at rest to create a K meson (m 3 c 2 = 494 Mev) and a lambda hyperon
(m t c 2 = 1115 Mev). Use conservation of energy and momentum, plus
relativistic kinematics, to find
(a) the kinetic energy in Mev of the incident pi meson at threshold for
production of K mesons, and compare this with the Q value of the
reaction;
(b) the kinetic energy of the pi meson in Mev in order to create K
mesons at 90° in the laboratory;
(c) the kinetic energy of K mesons emerging at 0° in the laboratory
when the kinetic energy of the pi meson is 20 per cent greater than in (b);
id) the kinetic energy of K mesons at 90° in the laboratory when the
incident pi meson has a kinetic energy of 1500 Mev.
12.4 It is a wellestablished fact that Newton's equation of motion
ma' = eE'
426 Classical Electrodynamics
holds for a small charged body of mass m and charge e in a coordinate
system K' where the body is momentarily at rest. Show that the Lorentz
force equation
dt \ c /
follows directly from the Lorentz transformation properties of accelera
tions and electromagnetic fields.
12.5 An alternative approach to the Lagrangian formalism for a relativistic
charged particle is to treat the 4vector of position x M and the 4velocity
UfZ = (yv, iyc) as Lagrangian coordinates. Then the EulerLagrange
equations have the obviously covariant form,
±i d ±\ d ±.=0
where L is a Lorentz invariant Lagrangian and t is the proper time.
(a) Show that
1 q ,
L =  mu^Up +  u^Ap
gives the correct relativistic equations of motion for a particle interacting
with an external field described by the 4vector potential A^.
(b) Define the canonical momenta and write out the Hamiltonian in
both covariant and spacetime form. The Hamiltonian is a Lorentz
invariant. What is its value ?
12.6 (a) Show from Hamilton's principle that Lagrangians which differ only
by a total time derivative of some function of the coordinates and time are
equivalent in the sense that they yield the same EulerLagrange equations
of motion.
(b) Show explicitly that the gauge transformation A^^Ap + (dKjdx^
of the potentials in the chargedparticle Lagrangian (12.75) merely
generates another equivalent Lagrangian.
12.7 A particle with mass m and charge e moves in a uniform, static, electric
field E .
(a) Solve for the velocity and position of the particle as explicit functions
of time, assuming that the initial velocity v was perpendicular to the
electric field.
(Jb) Eliminate the time to obtain the trajectory of the particle in space.
Discuss the shape of the path for short and long times (define "short"
and "long" times).
12.8 It is desired to make an E x B velocity selector with uniform, static,
crossed, electric and magnetic fields over a length L. If the entrance and
exit slit widths are Ax, discuss the interval Ah of velocities around the
mean value u = cEjB, which is transmitted by the device as a function of
the mass, the momentum or energy of the incident particles, the field
strengths, the length of the selector, and any other relevant variables.
Neglect fringing effects at the ends. Base your discussion on the practical
facts that L ~ few meters, ^x ~3xl0 4 volts/cm, Ax ~ 10^i to 10^ 2 cm,
u ~ 0.5 to 0.995c.
[Probs. 12] RelativisticP article Kinematics and Dynamics 427
12.9 A particle of mass m and charge e moves in the laboratory in crossed,
static, uniform, electric and magnetic fields. E is parallel to the x axis;
B is parallel to the y axis.
(a) For E < B make the necessary Lorentz transformation described
in Section 12.8 to obtain explicitly parametric equations for the particle's
trajectory.
(b) Repeat the calculation of (a) for E > B.
12.10 Static, uniform electric and magnetic fields, E and B, make an angle of 6
with respect to each other.
(a) By a suitable choice of axes, solve the force equation for the motion
of a particle of charge e and mass m in rectangular coordinates.
(b) For E and B parallel, show that with appropriate constants of
integration, etc., the parametric solution can be written
R
x = AR sin <j>, y = AR cos <$>, z =  Vl + A 2 cosh (pfi)
P
TO
ct =  Vl + A 2 sinh (p0)
P
where R = (mc^leB), p = (E/B), A is an arbitrary constant, and <f> is the
parameter [actually c/R times the proper time].
12.11 The magnetic field of the earth can be represented approximately by a
magnetic dipole of magnetic moment M = 8.1 x 10 25 gausscm 3 . Con
sider the motion of energetic electrons in the neighborhood of the earth
under the action of this dipole field (Van Allen electron belts).
{a) Show that the equation for a line of magnetic force is r = r sin 2 6,
where is the usual polar angle (colatitude) measured from the axis of the
dipole, and find an expression to the magnitude of B along any line of
force as a function of 0.
(b) A positively charged particle spirals around a line of force in the
equatorial plane with a gyration radius a and a mean radius R (a < R).
Show that the particle's azimuthal position (longitude) changes approxi
mately linearly in time according to
m
<f>(0=<f>o+^[jjco B (tt )
where (o B is the frequency of gyration at radius R.
(c) If, in addition to its circular motion of (b), the particle has a small
component of velocity parallel to the lines of force, show that it undergoes
small oscillations in around = tt/2 with a frequency Q = (3/ V2)(a/R)(o B .
Find the change in longitude per cycle of oscillation in latitude.
(d) For an electron of 10 Mev at a mean radius R = 3 x 10 9 cm, find
co B and a, and so determine how long it takes to drift once around the
earth and how long it takes to execute one cycle of oscillation in latitude.
Calculate these same quantities for an electron of 10 Kev at the same
radius.
12.12 A charged particle finds itself instantaneously in the equatorial plane of
the earth's magnetic field (assumed to be a dipole field) at a distance R
from the center of the earth. Its velocity vector at that instant makes an
angle a with the equatorial plane (y u /v x = tan a). Assuming that the
428 Classical Electrodynamics
particle spirals along the lines of force with a gyration radius a < i?, and
that the flux linked by the orbit is a constant of the motion, find an
equation for the maximum magnetic latitude A reached by the particle as
a function of the angle a. Plot a graph {not a sketch) of A versus a. Mark
parametrically along the curve the values of a for which a particle at
radius R in the equatorial plane will hit the earth (radius R ) for
R/R = 1.5, 2.0, 2.5, 3, 4, 6, 8, 10.
13
Collisions between
Charged Particles,
Energy Loss,
and Scattering
In this chapter collisions between swiftly moving, charged particles
are considered, with special emphasis on the exchange of energy between
collision partners and on the accompanying deflections from the incident
direction. A fast charged particle incident on matter makes collisions with
the atomic electrons and nuclei. If the particle is heavier than an electron
(mu or pi meson, K meson, proton, etc.), the collisions with electrons and
with nuclei have different consequences. The light electrons can take up
appreciable amounts of energy from the incident particle without causing
significant deflections, whereas the massive nuclei absorb very little energy
but because of their greater charge cause scattering of the incident particle.
Thus loss of energy by the incident particle occurs almost entirely in
collisions with electrons. The deflection of the particle from its incident
direction results, on the other hand, from essentially elastic collisions with
the atomic nuclei. The scattering is confined to rather small angles, so that
a heavy particle keeps a more or less straightline path while losing energy
until it nears the end of its range. For incident electrons both energy loss
and scattering occur in collisions with the atomic electrons. Consequently
the path is much less straight. After a short distance, electrons tend to
diffuse into the material, rather than go in a rectilinear path.
The subject of energy loss and scattering is an important one and is
discussed in several books* where numerical tables and graphs are
* See references at the end of the chapter.
429
430 Classical Electrodynamics
presented. Consequently our discussion will emphasize the physical ideas
involved, rather than the exact numerical formulas. Indeed, a full
quantummechanical treatment is needed to obtain exact results, even
though all the essential features are classical or semiclassical in origin.
The order of magnitude of the quantum effects are all easily derivable
from the uncertainty principle, as will be seen in what follows.
We will begin by considering the simple problem of energy transfer to a
free electron by a fast heavy particle. Then the effects of a binding force on
the electron are explored, and the classical Bohr formula for energy loss is
obtained. Quantum modifications and the effect of the polarization of
the medium are described, followed by a discussion of energy loss in an
electronic plasma. Then the elastic scattering of incident particles by
nuclei and multiple scattering are presented. Finally, a discussion is given
of the electrical resistivity of a plasma caused by screened Coulomb
collisions.
13.1 Energy Transfer in a Coulomb Collision
A swift particle of charge ze and mass M collides with an electron in an
atom. If the particle moves rapidly compared to the characteristic velocity
of the electron in its orbit, during the collision the electron can be treated
as free and initially at rest. As further approximations we will assume that
the momentum transfer Ap is sufficiently small that the incident particle
is essentially undeflected from its straightline path, and that the recoiling
electron does not move appreciably during the collision. Then to find the
energy transfer during the collision we need only calculate the momentum
impulse caused by the electric field of the incident particle at the position
of the electron. The particle's magnetic field is of negligible importance if
the electron is essentially at rest.
Figure 13.1 shows the geometry of the collision. The incident particle
has a velocity v and an energy E = yMc 2 . It passes the electron of charge e
and mass m < M at an impact parameter b. At the position of the elec
tron the fields of the incident particle are given by (11.118) with q = ze.
Only the transverse electric field E x has a nonvanishing time integral.
[Sect. 13.1] Collisions between Charged Particles 431
Consequently the momentum impulse A/? is in the transverse direction and
has the magnitude
Ap=\ eE x {t)dt = ^ (13.1)
J oo bv
It should be noted that Ap is independent of y, as discussed in Section
11.10 below Eq. (11.119). The energy transferred to the electron is
Am =m^(i\ (132)
2m mv 2 \b 2 /
The angular deflection of the incident particle is given by ~ A/?//?,
provided A/? < p. Thus, for small deflections,
~ — (13.3)
pub
This result can be compared with the wellknown exact expression for the
Rutherford scattering of a nonrelativistic particle of charge ze by a
Coulomb force field of charge z'e:
2 tan  = (13.4)
2 pu&
We see that for small angles the two expressions agree.*
The energy transfer &E(b) given by (13.2) has several interesting features.
It depends only on the charge and velocity of the incident particle, not on
its mass. It varies inversely as the square of the impact parameter so that
close collisions involve very large energy transfers. There is, of course, an
upper limit on the energy transfer, corresponding to a headon collision.
Our method of calculation is really valid only for large values of b. We
can obtain a lower limit b min on the impact parameter for which our
approximate calculation is valid by equating (13.2) to the maximum
allowable energy transfer (12.59):
AE(b min ) = A£ max = 2myV (13.5)
This yields the lower bound,
bmiQ = J? (13.6)
ymv
* Actually there is a question of reference frames in comparing (13.3) and (13.4).
Since (13.4) holds for a fixed center of force (or the CM system), we should compare it
with the result for the deflection of the light electron in the frame where the heavy
incident particle is at rest. Then (13.3) holds with/? ~ ymv as the electron momentum
in that frame. The reader may verify that (13.3) and (13.4) are also consistent in the
frame in which the electron is at rest by using (12.50) and (12.54) to transform angles
from the CM system to the laboratory.
432 Classical Electrodynamics
below which our approximate result (13.2) must be replaced by a more
exact expression which tends to (13.5) as b > 0. It can be shown (Problem
13.1) that a proper treatment gives the more accurate result,
A£(fr)~^( * ) (13.7)
mv 2 \fc min + bV
Equation (13.7) exhibits the proper limiting behavior as b »> and reduces
to (13.2) for b >b min .
The lower limit on b can be obtained by another argument. Equation
(13.2) was derived under the assumption that the electron did not move
appreciably during the collision. As long as the distance d it actually
moves is small compared to b, we may expect that (13.2) will be correct.
An estimate of d can be obtained by saying that Ap/2m is an average
velocity of the electron during the collision, and that the time of collision
is given by (11.120). Hence the distance traveled during the collision is of
the order of
Ap . ze 2
2m ymxr
As long as b > d, (13.2) should hold. This is exactly the condition
implied by (13.7).
At the other extreme of very distant collisions the approximate result
(13.2) for AE{b) is in error because of the binding of the atomic electrons.
We assumed that the electrons were free, whereas they are actually bound
in atoms. As long as the collision time (11.120) is short compared to the
orbital period of motion, it may be expected that the collision will be sudden
enough that the electron may be treated as free. If, on the other hand, the
collision time (11.120) is very long compared to the orbital period, the
electron will make many cycles of motion as the incident particle passes
slowly by and will be influenced adiabatically by the fields with no net
transfer of energy. The dividing point comes at impact parameter Z> max ,
where the collision time (1 1 . 120) and the orbital period are comparable. If
co is a characteristic atomic frequency of motion, this condition is
d~P xAt = — % = b min (13.8)
1
At(b ma .x) ~ —
co
or
b  y 
o max —
CO J
(13.9)
For impact parameters greater than & max it can be expected that the energy
transfer falls below (13.2), going rapidly to zero for b > b max .
The general behavior of AE(b) as a function of b is shown in Fig. 13.2.
The dotted curve represents the approximate form (13.2), while the solid
[Sect. 13.1]
Collisions between Charged Particles
433
Fig. 13.2 Energy transfer as a
function of impact parameter.
log 6
curve is a representation of the correct result. In the interval b m iu < b <
bm&x the energy transfer is given approximately by (13.2). But for impact
parameters outside that interval, the energy transfer is considerably less.
A fast particle passing through matter "sees" electrons at various
distances from its path. If there are N atoms per unit volume with Z
electrons per atom, the number of electrons located at impact parameters
between b and (b + db) in a thickness dx of matter is
dn = NZ 2irb db dx
(13.10)
To find the energy lost per unit distance by the incident particle we multiply
(13.10) by the energy transfer AE(b) and integrate over all impact para
meters. Thus the energy loss is
— = 2nNZ \AE(b)b db (13.11)
dx J
In view of the behavior of AE(b) shown in Fig. 13.2 we may use approxi
mation (13.2) and integrate between 6 m in and b mSbX . Then we find the result
dE
dx
_2„4 /*bmax 1
~4ttNZ— 9  9 bdb
mv 2 Jbmin b l
(13.12)
434
Classical Electrodynamics
or
dE zV
— ~4ttNZ— JnB
where
dx
B =
mv
y 2 mv z
ze 2 co
(13.13)
(13.14)
This approximate expression for the energy loss exhibits all the essential
features of the classical result due to Bohr (1915). The method of handling
the lower limit of integration in (13.12) is completely equivalent to using
(13.7) for AE(b). The cutoff at b = b ma , x is only approximate. Con
sequently B is uncertain by a factor of the order of unity. Because B
appears in the logarithm, this factor is of negligible importance numeri
cally. In any event, a proper treatment of binding effects is given in the
next section. Discussion of (13.13) as a function of energy and its com
parison with experiment will be deferred until Section 13.3.
13.2 Energy Transfer to a Harmonically Bound Charge
In order to justify the plausible value Z? max (13.9) of the impact parameter
which divides the Coulomb collisions for b < b max with the freeenergy
transfer (13.2) and essentially adiabatic collisions for b > b max with
negligible energy transfer, we consider the problem of the energy lost by a
massive charged particle with charge ze and velocity v passing a harmoni
cally bound charge of mass m and charge e. This will serve as a simplified
model for energy loss of particles passing through matter. As before, we
will assume that the massive particle is deflected only slightly in the
encounter so that its path can be approximated by a straight line. It passes
by the bound particle at an impact parameter b, measured from the origin
O of the binding force, as shown in Fig. 13.3. Since we are primarily
interested in large impact parameters where binding effects are important,
[Sect. 13.2] Collisions between Charged Particles 435
we may assume that the energy transfer is not large, that the motion of the
bound particle is nonrelativistic throughout the collision, and that its
initial and final amplitudes of oscillation about the origin O are small
compared to b. Then only the electric field of the incident particle need
be included in the force equation. Furthermore, its variation over the
position of the bound particle may be neglected, and its effective value can
be taken as that at the origin O. This is sometimes called the dipole approxi
mation, by analogy with the corresponding problem of absorption of
radiation.
With these approximations the force equation for the harmonically
bound charge can be written as
x + Tx + o> 2 x =  E(f) (13.15)
m
where E(0 is the electric field at O due to the charge ze, its components
being given by (11.118), co is the characteristic frequency of the binding,
and T is a small damping constant. The damping factor is not essential,
but it is present to at least some degree in actual physical systems and serves
to remove certain ambiguities which would arise in its absence. To solve
(13.15) we Fourieranalyze both E(?) and x(t):
x(0 = L (*" x(co)e i<ot dto (13.16)
■J2TT J oo
1 f °°
E(0 = ^= E((o)e itot dco (13.17)
■J2.1T J  oo
Since both x(0 and E(f) are real, the positive and negative frequency parts
of their transforms are related by
x(— a>) = x*(co)
V } V (13.18)
E(co) = E*(eo)
When the Fourier integral forms are substituted into the force equation,
we find
*(»> = £ ^^T ? (1319)
m co — icol — a>
With the known form of E(0 the Fourier amplitude E(co) can be deter
mined. Then x(0 can be found from (13.16), using (13.19). The problem
is solved, provided one can do the integrals.
The quantity of immediate interest is not the detailed motion of the
bound particle, but the energy transfer in the collision. This can be found
by considering the work done by the incident particle on the bound one.
436 Classical Electrodynamics
The rate of doing work is given by
f=jE. Jd V ( 13 .20)
Thus the total work done by the particle passing by is
AE=j CO dt\d 3 x'E>J (13.21)
The current density is J = ev d[x' — x(f)] for the bound charge. Con
sequently
AE = e\ \>Edt (13.22)
J — oo
where v = x, and in the dipole approximation E is the field of the incident
particle at the origin O. Using the Fourier representations (13.16) and
(13.17), as well as that for a delta function (2.52), and the reality con
ditions (13.18), the energy transfer can be written
AE = 2e Re iwx(co) • E*(co) dco (13.23)
If now the result (13.19) for x(co) is inserted, this becomes
AE= e \ E(o>) 2 1( °l d«> (13.24)
m Jo (w 2 — co 2 ) 2 + coT 2
For small T the integrand peaks sharply around a> = co in an approxi
mately Lorentzian line shape. Consequently the factor involving the
electric field can be approximated by its value at co = co . Then (13.24)
becomes
AE = — E(w ) 2 ' °° x%dx
m
co l 3?
o Lr 2
(13.25)
+ a; 2
The integral has the value tt/2, independent of coJT. Thus the energy
transfer is
AE = ™ 2 E(a> ) 2 (13.26)
Equation (13.26) is a very general result for energy transfer to a non
relativistic oscillator by an external electromagnetic field. In the present
application the field is produced by a passing charged particle. But a
pulse of radiation or any combination of external fields will serve as well.
For a particle with charge ze passing by the origin O at an impact
parameter b with a velocity v, the electromagnetic fields at the origin are
[Sect. 13.2]
Collisions between Charged Particles
437
given by (11.118) with q = ze. To illustrate the determination of the
Fourier transform (1 3. 1 7) we consider E x (t). Its transform E^w) is defined
to be
^^^rJ^+ywr (13  27)
By changing integration variable to x = yvtjb, (13.27) can be written as
iabxjyv
£l(«>) =
ze
r
dx
(13.28)
From a table of Fourier transforms* we find that the integral is propor
tional to a modified Bessel function of the order of unity [see (3.101)].
Thus
£i(C0)= 5£.(2fr^ Xi (^)i (13 . 29)
bv\TT/ Lyv \yv'
Similarly E 3 (t) given by (11.118) has the Fourier transform:
£3(<u )=_,il(2fr^K (^)l (13.30)
The energy transfer (13.26) to the harmonically bound charge can now
be evaluated explicitly. Using (13.29) and (13.30), we find
AE(b) =
2zV
&
where
y 2
0) n b
yv
(13.31)
(13.32)
The factor multiplying the square bracket is just the approximate result
(13.2). For small and large i, the limiting forms (3.103) and (3.104) show
that the square bracket in (13.31) has the limiting values:
[ ] =
1,
iK)i
!e
2
for £ < 1
for £ > 1
(13.33)
Since £ = bjb m&x , we see that for b < b m&x the energy transfer is essentially
the approximate result (13.2), while for b > Z> max it falls off exponentially
to zero. This justifies the qualitative arguments of the previous section on
the upper limit 6 m ax
* See, for example, Magnus and Oberhettinger, Chapter VIII, or Bateman Manuscript
Project, Tables of Integral Transforms, Vol. I, Chapters I— III.
438 Classical Electrodynamics
13.3 Classical and QuantumMechanical EnergyLoss Formulas
The energy transfer (13.31) to a harmonically bound charge can be used to
calculate a classical energy loss per unit length for a fast, heavy particle
passing through matter. We suppose that there are N atoms per unit
volume with Z electrons per atom. The Z electrons can be divided into
groups specified by the indexy, with^ electrons having the same harmonic
binding frequency co } . The number f j is called the oscillator strength of
they'th oscillator. The oscillator strengths satisfy the obvious sum rule,
2/; — z  Bv a trivial extension of the arguments leading to (13.11)
3
and (13.12) we find the energy loss to be
dE ^Z* f °°
— = 2ttN 2, /, kE^b db (13.34)
dX j J bmin
where A£/6) is given by (13.31) with I = cofi/yv, and a lower limit of
6min is specified, consistent with (13.7). No upper limit is necessary, since
(13.31) falls rapidly to zero for large b. The integral over the modified
Bessel functions can be done in closed form, leading to the result,
dE_ 47rNz 2 e*
dx mv 2 i
v 2
£minKi(! : min).K'o(£min)— — „ lmin(^ 1 2 (f min ) — K 2 (£ m in))
2c
(13.35)
where  min = wfiminlyv. In general, £ mi n < 1 . This means that the
limiting forms (3. 103) may be used to simplify (13.35). This final expression
for classical energy loss is
^ = 4nNZ —
dx mv 2
. ° 2c 2 J
(13.36)
where the argument of the logarithm is
_ \.\23yv 1.123yW
Be ~ 7TI — = i^^ — (13.37)
The average frequency (co) appearing in B c is a geometric mean defined
by
Z In (co) = 2L In co } (13.38)
3
The result (13.36)— (13.38) is that obtained by Bohr in his classic paper on
energy loss (1915). Our approximate expression (13.13) is in agreement
with (13.36) in all its essentials, since the added —v 2 /2c 2 is a small cor
rection even at high velocities.
[Sect. 13.3] Collisions between Charged Particles 439
Bohr's formula (13.36) gives a reasonable description of the energy loss
of relatively slow alpha particles and heavier nuclei. But for electrons,
mesons, protons, and even fast alphas, it overestimates the energy loss
considerably. The reason is that for the lighter particles quantum
mechanical modifications cause a breakdown of the classical result. The
important quantum effects are (1) discreteness of the possible energy
transfers, and (2) limitations due to the wave nature of the particles and
the uncertainty principle.
The problem of the discrete nature of the energy transfer can be illus
trated by calculating the classical energy transfer (13.2) at b ~ & m ax. This
is roughly the smallest energy transfer that is of importance in the energy
loss process. Assuming only one binding frequency a> for simplicity, we
find
AE(fr max ) ~  2 z 2 (ffi™o (1339)
where v = c/137 is the orbital velocity of an electron in the ground state
of hydrogen. Since Hco is of the order of the ionization potential of the
atom, we see that for a fast particle (v > v ) the classical energy transfer
(13.39) is very small compared to the ionization potential, or even to the
smallest excitation energy in the atom. But we know that energy must be
transferred in definite quantum jumps. A tiny amount of energy like (1 3.39)
simply cannot be absorbed by the atom. We conclude that our classical
calculation fails in this domain. We might argue that only if our classical
formula (13.2) gives an energy transfer large compared to typical atomic
excitation energies would we expect it to be correct. This would set quite
a different upper limit on the impact parameters. Fortunately the classical
result can be applied in a statistical sense if we reinterpret its meaning.
Quantum considerations show that the classical result of the transfer of a
small amount of energy in every collision is incorrect. But if we consider
a large number of collisions, we find that on the average a small amount
of energy is transferred. It is not transferred in every collision, however.
In most collisions no energy is transferred. But in a few collisions an
appreciable excitation occurs, yielding a small average value over many
collisions. In this statistical sense the quantum mechanism for discrete
energy transfers and the classical process with a continuum of possible
energy transfers can be reconciled. The detailed numerical agreement
stems from the quantummechanical definitions of the oscillator strengths
fj and resonant frequencies co>.
The other important quantum modification arises from the wave nature
of the particles. The uncertainty principle sets certain limits on the range
of validity of classical orbit considerations. If we try to construct wave
440 Classical Electrodynamics
packets to give approximate meaning to a classical trajectory, we know
that the path can be defined only to within an uncertainty Ax > hjp. For
impact parameters b less than this uncertainty, classical concepts fail.
Since the wave nature of the particles implies a smearing out in some sense
over distances of the order of Ax, we anticipate that the correct quantum
mechanical energy loss will correspond to much smaller energy transfers
than given by (13.2) for b < Ax. Thus Ax ~ hip is a quantum analog of
the minimum impact parameter (13.6). In the collision of two particles
each one has a wave nature. For a given relative velocity the limiting
uncertainty will come from the lighter of the two. For a heavy incident
particle colliding with an electron, the momentum of the electron in the
coordinate frame where the incident particle is at rest (almost the CM
frame) is p' = ymv, where m is the mass of the electron. Therefore the
quantummechanical minimum impact parameter is
h
Cn = (13.40)
ymv
For electrons incident on electrons we must take more care and consider
the CM momentum (12.34) for equal masses. Then for electrons we obtain
the minimum impact parameter,
(«) \ ^ I ^
[^min J electrons = / (13.41)
mc ^ y — 1
In a given situation the larger of the two minimum impact parameters
(13.6) and (13.40) must be used to define argument B (13.14) of the
logarithm in dE/dx. The ratio of the classical to quantum value of b min is
ze 2
ri = — (13.42)
nv
Ift]> 1, the classical Bohr formula must be used. We see that this occurs
for slow, highly charged, incident particles, in accord with observation.
If r\ < 1, the quantum minimum impact parameter is larger than the
classical one. Then quantum modifications appear in the energyloss
formula. The argument of the logarithm in (13.13) becomes
„ bmax „ y 2 mv 2
B « = ^r =r i B = r7 (13.43)
Equation (13.13) with the quantummechanical B q (13.43) in the logarithm
is a good approximation to a quantumtheoretical result of Bethe (1930).
Bethe's formula, including the effects of close collisions, is
dE z 2 e*
^ = 4ttNZ —
dx mv 2
ln 1 2 r 2mv ''
h{<o)
J c\
(13.44)
[Sect. 13.3]
Collisions between Charged Particles
441
Apart from the small correction term v 2 \c 2 and a factor of 2 in the
argument of the logarithm, this is just our approximate expression.
For electrons the quantum effects embodied in (13.41) lead to a modified
quantummechanical argument for the logarithm:
B d ^ (y  1)
y A mc 2
j y + 1 mc 2
2 hico)^ ~^2 h(a>)
(13.45)
where the last expression is valid at high energies. Even though there are
other quantum effects for electrons, such as spin and exchange effects, the
dominant modifications are included in (13.45).
The general behavior of both the classical and quantummechanical
energyloss formulas is shown in Fig. 13.4. At low energies, the main
energy variation is as v~ 2 , since the logarithm changes slowly. But at high
energies where v ► c the variation is upwards again, going as In y for
y > 1. Bethe's formula is in good agreement with experiment for all fast
particles with r\ < 1, provided the energy is not too high (see the next
section).
It is worth while to note the physical origins of the two powers of y
which appear in B Q (13.43). One power of y comes from the increase of
the maximum energy (13.5) which can be transferred in a headon collision.
The other power comes from the relativistic change in shape of the electro
magnetic fields (11.118) of a fast particle with the consequent shortening
of the collision time (11.120) and increase of fcmax (13.9). The fields are
effective in transferring energy at larger distances for a relativistic particle
than for a nonrelativistic one.
Sometimes it is of interest to know the energy loss per unit distance due
to collisions in which less than some definite amount e of energy is trans
ferred per collision. In photographic emulsions, for example, ejected
Fig. 13.4 Energy loss as a
function of kinetic energy.
0.01 0.1
1
10
HT
10" 5
10 4
<T 1)= ^
442 Classical Electrodynamics
electrons of more than about 10 Kev energy have a range greater than the
average linear dimensions of the silver bromide grains. Consequently the
energy dissipated in blackening of the grains corresponds to collisions
where the energy transfer is less than about 10 Kev. Classically, the
desired energyloss formula can be obtained from the Bohr formula (13.35)
with a minimum impact parameter b min (€) chosen so that (13.2) is equal
to e. Thus
2ze 2
&min(e) = — — T (13.46)
u(2rae)
This leads to a formula of the form of (13.36), but with an argument in
the logarithm,
B/O 'T* 2 "*' (1347)
Since quantummechanical energyloss formulas are obtained from clas
sical ones by the replacement [see (13.43)],
ze 2
B a = rjB c = — B c (13.48)
nv
we expect that the quantummechanical formula for energyloss per unit
distance due to collisions with energy transfer less than e will be
(13.49)
dF 2 V r Ji 2
f*( € ) = 4 7 rNZ^ 2 \nB Q (e)^
ax mtr L 2c 
where , ,
BJAXV&& (13.50)
n{co)
The constant A is a numerical factor of the order of unity that cannot be
determined without detailed quantummechanical calculations. Bethe's
calculations (1930) give the value A = 1. The quantummechanical 5 g (e)
can be written as
where 6 max is given by (13.9), and the minimum impact parameter is
b$ n (e)~—~——^ (13.52)
The implication of this formula is that the classical trajectory must be ill
defined by an amount at least as great as (13.52) in order that the uncertainty
in transverse momentum Ap be less than the momentum transfer in the
collision. Otherwise we would be unable to be certain that an energy
transfer of less than e had actually occurred. Hence (13.52) forms a
natural quantummechanical lower limit on the classical orbit picture in
this case.
[Sect. 13.4] Collisions between Charged Particles 443
13.4 Density Effect in Collision Energy Loss
For particles which are not too relativistic the observed energy loss is
given accurately by (13.44) [or by (13.36) if rj > 1] for all kinds of particles
in all types of media. For ultrarelativistic particles, however, the observed
energy loss is less than predicted by (13.44), especially for dense substances.
In terms of Fig. 13.4 of (dE/dx), the observed energy loss increases beyond
the minimum with a slope of roughly onehalf that of the theoretical curve,
corresponding to only one power of y in the argument of the logarithm
in (13.44) instead of two. In photographic emulsions the energy loss, as
measured from grain densities, barely increases above the minimum to a
plateau extending to the highest known energies. This again corresponds
to a reduction of one power of y, this time in ^(e) (13.50).
This reduction in energy loss, knbwn as the density effect, was first
treated theoretically by Fermi (1940). In our discussion so far we have
tacitly made one assumption that is not valid in dense substances. We
have assumed that it is legitimate to calculate the effect of the incident
particle's fields on one electron in one atom at a time, and then sum up
incoherently the energy transfers to all the electrons in all the atoms with
b min < * < &max. Now & max is very>rge compared to atomic dimensions,
especially for large y. Consequently in dense media there are many atoms
lying between the incident particle's trajectory and the typical atom in
question if b is comparable to £ ma x. These atoms, influenced themselves
by the fast particle's fields, will produce perturbing fields at the chosen
atom's position, modifying its response to the fields of the fast particle.
Said in another way, in dense media the dielectric polarization of the
material alters the particle's fields from their freespace values to those
characteristic of macroscopic fields in a dielectric. This modification of
the fields due to polarization of the medium must be taken into account
in calculating the energy transferred in distant collisions. For close
collisions the incident particle interacts with only one atom at a time. Then
the freeparticle calculation without polarization effects will apply. The
dividing impact parameter between close and distant collisions is of the
order of atomic dimensions. Since the joining of two logarithms is involved
in calculating the sum, the dividing value of b need not be specified with
great precision.
We will determine the energy loss in distant collisions (b > a), assuming
that the fields in the medium can be calculated in the continuum approxi
mation of a macroscopic dielectric constant e(co). If a is of the order of
atomic dimensions, this approximation will not be good for the closest of
the distant collisions, but will be valid for the great bulk of the collisions.
444
Classical Electrodynamics
The problem of finding the electric field in the medium due to the incident
fast particle moving with constant velocity can be solved most readily by
Fourier transforms. If the potentials A^x) and source density J (x) are
transformed in space and time according to the general rule,
x —iwt
(13.53)
(13.54)
F(x, t) = — 2 \d*k \dco F(k, co)e ik '
then the transformed wave equations become
m 2 1 4tt "•
k 2 e(co) 0>(k, co) = ^ p(k, co)
L c L J e(co)
k 2 ^ e(co) 1 A(k, co) = — J(k, co)
L c 2 J c
The dielectric constant e(co) appears characteristically in positions dictated
by the presence of D in Maxwell's equations. The Fourier transforms of
p(x, t) = zed(x  Yt))
and (13.55)
J(x, = \ P (x, t) J
are readily found to be
ze
/>(k, co) = — d(co — k • v)
Itt
(13.56)
J(k, a>) = vp(k, oi)
From (13.54) we see that the Fourier transforms of the potentials are
2ze d(co — k • v)
0(k, co) =
and
e(co) , 2 co 2 .
c 2
A(k, co) = e(oj)  0(k, co)
(13.57)
From the definitions of the electromagnetic fields in terms of the potentials
we obtain their Fourier transforms :
_,,, x .((0€(co)y . ,^,,
E(k, co) = / — L  '  — k )^>(k, co)
\ c c
B(k, co) = ie(co) k x  <D(k, co)
c
(13.58)
In calculating the energy loss it is apparent from (13.23) that we want
the Fourier transform in time of the electromagnetic fields at a perpen
dicular distance b from the path of the particle moving along the z axis.
Thus the required electric field is
E(co) = — Jyr d 3 k E(k, co) e ibkl
(2tt) 2 J
(13.59)
[Sect. 13.4] Collisions between Charged Particles 445
where the observation point has coordinates (b, 0, 0). To illustrate the
determination of E(w) we consider the calculation of E 3 (w), the component
of E parallel to v. Inserting the explicit forms from (13.57) and (13.58), we
obtain
«„) = ^ > U , (^  is )ii^> (13.60)
/c 2 ^<o>)
c 2
The integral over d£: 3 can be done immediately. Then
(13.61)
where 22 2
A 2 = «» _ £L e(co) = ^ (1  £ 2 <co)) (13.62)
y 2 c 2 U
The integral over dk % has the value 77/(A 2 + ^i 2 ) 1 ^, so that E z {oS) can be
written
«>^&')£<F^ (13  63)
The remaining integral is of the same general structure as (13.28). The
result is
£ 3(K1 ) =  J22L (2) M (J  /)•) K (A!>) (13.64)
where the square root of (13.62) is chosen so that A lies in the fourth
quadrant. A similar calculation yields the other fields :
X
V \tt/
K&by
6(0) (13.65)
B 2 (co) = e(co) / S£ 1 (co) >
In the limit e(co) > 1 it is easily seen that fields (13.64) and (13.65) reduce
to the earlier results (13.30) and (13.29).
To find the energy transferred to the atom at impact parameter b we
merely write down the generalization of (13.23):
J"QO
 io>x/co) . E*(o>) dm (13.66)
where ^{co) is the amplitude of the yth type of electron in the atom.
Rather than use (13.19) for x/cu) we express the sum of dipole moments
in terms of the molecular polarizability and so the dielectric constant:
e Y/,x» = — (e(co)  l)E(co) (13.67)
■£< 4ttN
446 Classical Electrodynamics
where TV is the number of atoms per unit volume. Then the energy transfer
can be written
1 f 00
AE(b) = —  Re ia> e(co) E(co) 2 dco (13.68)
zttN Jo
The energy loss per unit distance in collisions with impact parameter
b > a is evidently
(fL = HT A£(fe > wi < i3  69 >
If fields (13.64) and (13.65) are inserted into (13.68) and (13.69), we find,
after some calculation, the expression due to Fermi,
\t\ =  ( nr Re  ™&°Ki&*a)KMa)(rp\dm (13.70)
\dx/b>a 77 v* Jo Ve(ft)) /
where X is given by (13.62). This result can be obtained more elegantly by
calculating the electromagnetic energy radiated through a cylinder of
radius a around the path of the incident particle. By conservation of
energy this is the energy lost per unit time by the incident particle. Thus
(dE\ IdE c r „
— I =T=T— 2naB 2 E 3 dz (13.71)
\dx/b>a v dt ^vJoo ' v '
The integral over dz at one instant of time is equivalent to an integral at
one point on the cylinder over all time. Using dz = v dt, we have
(Si>.f r.^^ * (i372)
In the standard way this can be converted into a frequency integral,
— ) = ca Re B 2 *(<o)E 3 (o)) da> (13.73)
\dx/b>a J
With fields (13.64) and (13.65) this gives the Fermi result (13.70).
The Fermi expression (13.70) bears little resemblance to our previous
results for energy loss, such as (13,35). But under conditions where
polarization effects are unimportant it yields the same results as before.
For example, for nonrelativistic particles (/ff < 1) it is clear from (13.62)
that X ^ co/v, independent of e(o>). Then in (13.70) the modified Bessel
functions are real. Only the imaginary part of l/e(co) contributes to the
integral. If we neglect the Lorentz polarization correction (4.67) to the
internal field at an atom, the dielectric constant can be written
«■>* 1+^2 . S i , r (13.74)
[Sect. 13.4] Collisions between Charged Particles 447
where we have used the dipole moment expression (13.19). Assuming
that the second term is small, the imaginary part of l/e(co) can be readily
calculated and substituted into (13.70). Then the integral over dco can be
performed in the same approximation as used in (13.24)— (13.26) to yield
the nonrelativistic form of (13.35). If the departure of A from cojyv is
neglected, but no other approximations are made, then (13.70) yields
precisely the Bohr result (13.35).
The density effect evidently comes from the presence of complex
arguments in the modified Bessel functions, corresponding to taking into
account e(co) in (13.62). Since e(co) there is multiplied by /5 2 , it is clear that
the density effect can be really important only at high energies. The
detailed calculations for all energies with some explicit expression such as
(13.74) for e(co) are quite complicated and not particularly informative.
We will content ourselves with the extreme relativistic limit (/? ~ 1).
Furthermore, since the important frequencies in the integral over da> are
optical frequencies and the radius a is of the order of atomic dimensions,
\Xa\ ^ ' (coa/c) < 1 . Consequently we can approximate the Bessel functions
by their small argument limits (3.103). Then in the relativistic limit the
Fermi expression (13.70) is
\dx/b>a 77 c 2 Jo Wco) /
[.(!
(CO)
?V\ 1 1
dw (13.75)
123c \ 1
In (1 — e(a>))
coa J 2
It is worth while right here to point out that the argument of the second
logarithm is actually [1 — /# 2 e(co)]. In the limit e = 1, this log term gives
a factor y in the combined logarithm, corresponding to the old result
(13.36). Provided e(co) =£ 1, we can write this factor as [1 — €(co)],
thereby removing one power of y from the logarithm, in agreement with
experiment.
The integral in (13.75) with e(co) given by (13.74) can be performed most
easily by using Cauchy's theorem to change the integral over positive real
o> to one over positive imaginary co, minus one over a quarter circle at
infinity. The integral along the imaginary axis gives no contribution. Pro
vided the Tj in (13.74) are assumed constant, the result of the integration
over the quarter circle can be written in the simple form:
\dx/b>a c £ \ aco p I
where co„ is the electronic plasma frequency
, 4nNZe 2
m
(13.77)
448 Classical Electrodynamics
The corresponding relativistic expression without the density effect is,
from (13.36),
(dE\ = (zefail T /U23yc\ _ l\
\dx/b>a c 2 L \ a(co) J 2J
We see that the density effect produces a simplification in that the
asymptotic energy loss no longer depends on the details of atomic
structure through (a>) (13.38), but only on the number of electrons per unit
volume through co p . Two substances having very different atomic struc
tures will produce the same energy loss for ultrarelativistic particles pro
vided their densities are such that the density of electrons is the same in each.
Since there are numerous calculated curves of energy loss based on
Bethe's formula (13.44), it is often convenient to tabulate the decrease in
energy loss due to the density effect. This is just the difference between
(13.78) and (13.76):
P+i \dxf c 2 L \<o>>/ 2
(13.79)
For photographic emulsions, the relevant energy loss is given by (13.49)
and (13.50) with e^ 10 Kev. With the density correction applied, this
becomes constant at high energies with the value,
2_ 2
dE(e) (zeYco P
dx 2c 2
/2mc 2 e \
U 2 co 2 /
In \^i) (13.80)
For silver bromide, hco p ^ 48 ev. Then for singly charged particles (13.80),
divided by the density, has the value of approximately 1 .02 Mevcm 2 /gm.
This energy loss is in good agreement with experiment, and corresponds
to an increase above the minimum value of less than 10 per cent. Figure
13.5 shows total energy loss and loss from transfers of less than 10 Kev
for a typical substance. The dotted curve is the Bethe curve for total
energy loss without correction for density effect.
There is an interesting connection between the Fermi expression (13.70)
for energy loss and the emission of Cherenkov radiation. Equation (13.70)
represents energy transferred to the medium at distances greater than a.
If we let a > oo we can find out whether any of the energy escapes to
infinity. Such energy would be properly described as radiation. For
a > oo, the asymptotic forms (3.104) of the ^functions can be used. Then
(13.70) takes the form:
lira («) _ & Re r to (j_ _ ^(m<— >» *, (13.8D
a— oo \dx/b>a \) Jq \e(<X>) I \ A. /
[Sect. 13.4]
Collisions between Charged Particles
449
313
< 10 kev
0.1
10 10 2 10 3
(T  1) ^
10 4
Fig. 13.5 Energy loss, including the density effect. The dotted curve is the total energy
loss without density correction. The solid curves have the density effect incorporated,
the upper one being the total energy loss and the lower one the energy loss due to
individual energy transfers of less than 10 Kev.
If A has a real part, the exponential factor causes the energy loss to go
rapidly to zero at large distances. From (13.62) it is evident that this will
always occur if the medium is absorbent, since then e(o>) has a positive
imaginary part. But if e(w) is real, A can be pure imaginary for certain co.
This occurs whenever ft* > l/e(co), i.e., whenever the velocity of the part
icle is greater than the phase velocity of light in the medium. This is the
criterion for Cherenkov radiation. For such frequencies, A = — * A.
Then the exponential equals unity, and we find
Um (M) _6£f Jt _J_ ) dm (13 .82)
Since this expression is independent of the cylinder radius a, it represents
true radiation. It is just the FrankTamm (1937) result for the total energy
per unit distance emitted as Cherenkov radiation. A more detailed
discussion of Cherenkov radiation as a radiative process will be given in
Section 14.9.
For media in which the density effect is an important feature of the
energyloss process the absorption is almost always sufficiently great that
the incipient Cherenkov radiation is absorbed very close to the path of the
particle.
450 Classical Electrodynamics
13.5 Energy Loss in an Electronic Plasma
The loss of energy by a nonrelativistic particle passing through a plasma
can be treated in a manner similar to the density effect for a relativistic
particle. As was discussed in Section 10.10, the length scale in a plasma is
divided into two regions. For dimensions large compared to the Debye
screening distance k D ~ x (10.106), the plasma acts as a continuous medium
in which the charged particles participate in collective behavior such as
plasma oscillations. For dimensions small compared to k D ~ x , individual
particle behavior dominates and the particles interact by the twobody
screened potential (10.1 13). This means that in calculating energy loss the
Debye screening distance plays the same role here as the atomic dimension
a played in the densityeffect calculation. For close collisions collective
effects can be ignored, and the twobody screened potential can be used to
evaluate this contribution to the energy loss. This is left as an exercise for
the reader (Problem 13.3). For the distant collisions at impact parameters
bk D > 1 the collective effects can be calculated by utilizing Fermi's
formula (13.70) with an appropriate dielectric constant for a plasma. The
loss in distant collisions corresponds to the excitation of plasma oscillations
in the medium.
For a nonrelativistic particle (13.70) yields the following expression for
the energy loss to distances b > k D ~ x :
*l\ ~ 2 (* g ) 2
dx'k s b>l 77 V 2
Jo €(oj)\k D v \k D v/ \k D v/ .
dm (13.83)
Since the important frequencies in the integral turn out to be m ~ m v , the
relevant argument of the Bessel functions is
m v _ {u )
k n v v
(13.84)
For particles incident with velocities v less than thermal velocities this
argument is large compared to unity. Because of the exponential falloff
of the Bessel functions for large argument, the energy loss in exciting
plasma oscillations by such particles is negligible. Whatever energy is lost
is in close binary collisions. If the velocity is comparable with or greater
than thermal speeds, then the particle can lose appreciable amounts of
energy in exciting collective oscillations. It is evident that this energy of
oscillation is deposited in the neighborhood of the path of the particle,
out to distances of the order of (v/(u 2 y A ) kjf x .
[Sect. 13.6] Collisions between Charged Particles 451
For a particle moving rapidly compared to thermal speeds we may use
the familiar small argument forms for the modified Bessel functions. Then
(13.83) becomes
(f) = ^r Re /^,/u23M) dM (1385)
\ax>ic D b>i it v Jo \e(ft>)/ \ co 1
We shall take the simple dielectric constant (7.93), augmented by some
damping:
<°* = 1 ~ a?' T, ( 13  86 )
co 1 + icoT
The damping constant Y will be assumed small compared to co v . The
necessary combination,
Re (fl) = < rs %+ v> (I3 ' 87)
\e(a))/ (co — co v ) + co L
has the standard resonant character seen in (13.24), for example. In the
limit r < co v the integral in (13.85) leads to the simple result,
Ui\ ^ a/la (l^hnE) (1 3.88)
This can be combined with the results of Problem 13.3 to give an expression
for the total energy loss of a particle passing through a plasma. The
presence of co v in the logarithm implies that the energy losses occur in
quantum jumps of Hco p , in the same way as the mean frequency (co) in
(13.44) is indicative of the typical quantum jumps in atoms. Electrons
passing through thin metal foils show this discreteness in their energy loss.
The phenomenon can be used to determine the effective plasma frequency
in metals.
13.6 Elastic Scattering of Fast Particles by Atoms
In the preceding sections we have been concerned with the energy loss
of particles passing through matter. In these considerations it was assumed
that the trajectory of the particle was a straight line. Actually this
approximation is not rigorously true. As was discussed in Section 13.1,
any momentum transfer between collision partners leads to a deflection in
angle. In the introductory remarks at the beginning of the chapter it was
pointed out that collisions with electrons determine the energy loss,
whereas collisions with atoms determine the scattering. If the screening of
the nuclear Coulomb field by the atomic electrons is neglected, a fast
452 Classical Electrodynamics
particle of momentum/? = yMv and charge ze, passing a heavy nucleus of
charge Ze at impact parameter b, will suffer an angular deflection,
2z7e 2
~ ±f±L (13.89)
pvb
according to (13.3).
The differentia] scattering cross section da/dQ. (with dimensions of area
per unit solid angle per atom) is defined by the relation,
nbdbd<f> = n — sin 6 d6 d<f> (13.90)
dQ.
where n is the number of particles incident on the atom per unit area per
unit time. The lefthand side of (1 3.90) is the number of particles per unit
time incident at azimuthal angles between cf> and ((f> + d(f>) and impact
parameters between b and (b + db). The righthand side is the number of
scattered particles per unit time emerging at polar angles (6, (f>) in the
element of solid angle d£l = sin 6 dd d<f>. Equation (13.90) is merely a
statement of conservation of particles, since b and 6 are functionally
related. The classical differential scattering cross section can therefore be
written.
da
dQ. sin
db
dd
(13.91)
The absolute value sign is put on, since db and dd can in general have
opposite signs, but the cross section is by definition positive definite. If b
is a multiplevalued function of 6, then the different contributions must be
added in (13.91).
With relation (13.89) between b and 6 we find the smallangle nuclear
Rutherford scattering cross section per atom,
*^(™£\ % L (13 .92)
dQ \ pv I 4
We note that the Z electrons in each atom give a contribution Z 1 times
the nuclear one. Hence the electrons can be ignored, except for their
screening action. The smallangle Rutherford law (13.92) for nuclear
scattering is found to be true quantum mechanically, independent of the
spin nature of the incident particles. At wide angles spin effects enter, but
for nonrelativistic particles the classical Rutherford formula,
^ = (^£!Y cosec ^ (13.93)
dQ \2MvV 2
which follows from (13.4), holds quantum mechanically as well.
[Sect. 13.6] Collisions between Charged Particles 453
Since most of the scattering occurs for 6 < 1, and even at 6 = tt\2 the
smallangle result (13.92) is within 30 per cent of the Rutherford expression,
it is sufficiently accurate to employ (13.92) at all angles for which the
unscreened point Coulombfield description is valid.
Departures from the point Coulombfield approximation come at large
and small angles, corresponding to small and large impact parameters. At
large b the screening effects of the atomic electrons cause the potential to
fall off more rapidly than (1/r). On the FermiThomas model the potential
can be approximated roughly by the form :
V(r) ~ ^ exp (r/a) (13.94)
r
where the atomic radius a is
a^l.4«oZ 1/3 (13.95)
The length a = h 2 /me 2 is the hydrogenic Bohr radius. For impact param
eters of the order of, or greater than, a the rapid decrease of the potential
(13.94) will cause the scattering angle to vanish much more rapidly with
increasing b than is given by (13.89). This implies that the scattering cross
section will flatten off at small angles to a finite value at 6 = 0, rather than
increasing as 4 . A simple calculation with a cutoff Coulomb potential
shows that the cross section has the general form:
f^f^T l (13 96)
dQ. \ P V / (0 2 + 0Ln) 2
where min is a cutoff angle. The minimum angle m in below which the
cross section departs appreciably from the simple result (13.92) can be
determined either classically or quantum mechanically. As with b m i n in
the energyloss calculations, the larger of the two angles is the correct one
to employ. Classically min can be estimated by putting b = a in (13.89).
This gives
0^n~— (13.97)
pva
Quantum mechanically, the finite size of the scatterer implies that the
approximately classical trajectory must be localized to within Aa; < a;
the incident particle must have a minimum uncertainty in transverse
momentum Ap ^ h/a. For collisions in which the momentum transfer
(13.1) is large compared to H/a the classical Rutherford formula will apply.
But for smaller momentum transfers we expect the quantummechanical
smearing out to flatten off the cross section. This leads to a quantum
mechanical d m i n :
( X ^ — (13.98)
pa
454 Classical Electrodynamics
We note that the ratio of the classical to quantummechanical angles
0min is Zze 2 /hv in agreement with the ratio (13.42) of the classical and
quantum values of b m in. For fast particles in all but the highest Z sub
stances (Zze 2 lhv) is less than unity. Then the quantum value (13.98) will
be used for min . With value (13.95) for the screening radius a, (13.98)
becomes
ft^H (13.99)
192 \ p I
where p is the incident momentum (p = yMv), and m is the electronic
mass.
At comparatively large angles the cross section departs from (13.92)
because of the finite size of the nucleus. For electrons and mu mesons the
influence of nuclear size is a purely electromagnetic effect, but for pi mesons
protons, etc., there are specific effects of a nuclearforce nature as well.
Since the gross overall effect is to lower the cross section below that
predicted by (13.92) for whatever reason, we will consider only the
electromagnetic aspect. The charge distribution of the atomic nucleus can
be crudely approximated by a uniform volume distribution inside a sphere
of radius R, falling rapidly to zero outside R. This means that the electro
static potential inside the nucleus is not 1/r, but rather parabolic in shape
with a finite value at r = 0:
2 R \ 3R 2 '
V(r) =
zZe 2
(13.100)
for r > R
It is a peculiarity of the pointcharge Coulomb field that the quantum
mechanical cross section is the classical Rutherford formula. Thus for a
point nucleus there is no need to consider a division of the angular region
into angles corresponding to impact parameters less than, or greater than,
the quantummechanical impact parameter b<£ in (13.40). For a nucleus
of finite size, however, the de Broglie wavelength of the incident particle
does enter. When we consider wave packets incident on the relatively
constant (inside r = R) potential (13.100), there will be appreciable
departures from the simple formula (13.92). The situation is quite
analogous to the diffraction of waves by a spherical object, considered in
Chapter 9. The scattering is all confined to angles less than ~(X/R), where
X is the wavelength (divided by 2tt) of the waves involved. For wider
angles the wavelets from different parts of the scatterer interfere, causing
a rapid decrease in the scattering or perhaps subsidiary maxima and
[Sect. 13.6]
Collisions between Charged Particles
455
Fig. 13.6 Atomic scattering,
including effects of electronic
screening at small angles and
finite nuclear size at large angles.
Iog0
minima. Since the particle wavelength is X = h/p, the maximum scattering
angle, beyond which the scattering cross section falls significantly below the
0~ 4 law, is
pR
(13.101)
Using the simple estimate R ~ % (e 2 /mc 2 ) A 1A = \AA V * x 10 13 cm, this
has the numerical value,
/ / /I / wi /> v
(13.102)
274
mc
A" \p
We note that, for all values of Z and A, 6 ma , x > m i n . If the incident
momentum is so small that max > 1 , the nuclear size has no appreciable
effect on the scattering. For an aluminum target max = 1 when p ~ 50
Mev/c, corresponding to ^ '50 Mev, 12 Mev, and 1.3 Mev kinetic energies
for electrons, mu mesons, and protons, respectively. Only at higher
energies than these are nuclearsize effects important in the scattering. At
this momentum value d$ in ~ 10 4 radian.
The general behavior of the cross section is shown in Fig. 13.6. The
dotted curve is the smallangle Rutherford approximation (13.92), while
the solid curve shows the qualitative behavior of the cross section, includ
ing screening and finite nuclear size. The total scattering cross section can
be obtained by integrating (13.96) over all solid angle:
• fl^fl^^o /2zZe 2 \ 2 f°° Odd
sin 6 dd d(f> ^ 2tt\
(da .
= — sn
J dQ.
X
This yields
pv
! )X
(2zZe**
O ~ 771
\ pv
1
J2.
'mm
= TTd
V^min
JTzZfi
\ Hv /
+ 2 ) 2
(13.103)
(13.104)
456 Classical Electrodynamics
where the final form is obtained by using 0£(j n (13.98). It shows that at
high velocities the total cross section can be far smaller than the classical
value of geometrical area ira 2 .
13.7 Mean Square Angle of Scattering and the Angular Distribution of
Multiple Scattering
Rutherford scattering is confined to very small angles even for a point
Coulomb field, and for fast particles d max is small compared to unity. Thus
there is a very large probability for smallangle scattering. A particle
traversing a finite thickness of matter will undergo very many smallangle
deflections and will generally emerge at a small angle which is the cumu
lative statistical superposition of a large number of deflections. Only
rarely will the particle be deflected through a large angle; since these
events are infrequent, such a particle will have made only one such
collision. This circumstance allows us to divide the angular range into
two regions — one region at comparatively large angles which contains only
the single scatterings, and one region at very small angles which contains
the multiple or compound scatterings. The complete distribution in angle
can be approximated by considering the two regions separately. The
intermediate region of socalled plural scattering must allow a smooth
transition from small to large angles.
The important quantity in the multiplescattering region, where there
is a large succession of smallangle deflections symmetrically distributed
about the incident direction, is the mean square angle for a single scattering.
This is defined by
(S 2 ) = J r / Q (13.105)
J da
With the approximations of Section 13.6 we obtain
<0 2 > = 20U In (^) (13.106)
^ Oram'
If the quantum value (13.99) of m in is used along with max (13.102), then
(13.106) has the numerical form:
<0 2 > ~ 40f nin In (210Z*) (13.107)
If nuclear size is unimportant (generally only of interest for electrons, and
perhaps other particles at very low energies), max should be put equal to
[Sect. 13.7]
Collisions between Charged Particles
457
Fig. 13.7
unity in (13.106). Then the argument of the logarithm in (13.107) becomes
/192 p V*
instead of (210Z~^).
It is often desirable to use the projected angle of scattering d', the
projection being made on some convenient plane such as the plane of a
photographic emulsion or a bubble chamber, as shown in Fig. 13.7. For
small angles it is easy to show that
<0' 2 > = i<0 2 >
(13.108)
In each collision the angular deflections obey the Rutherford formula
(13.92) suitably cut off at min and m ax, with average value zero (when
viewed relative to the forward direction, or as a projected angle) and mean
square angle (0 2 ) given by (13.106). Since the successive collisions are
independent events, the centrallimit theorem of statistics can be used to
show that for a large number n of such collisions the distribution in angle
will be approximately Gaussian around the forward direction with a mean
square angle <© 2 ) = n (0 2 ). The number of collisions occurring as the
particle traverses a thickness / of material containing N atoms per unit
VOlUme iS tj y 2\2 ,
n = Not~TrN\±^)! (13.109)
\ BV / 0min
This means that the mean square angle of the Gaussian is
<0 2 > ~ 2ttN
mn
(13.110)
P M (d') dd' = p4== exp ( ) dd' (13.112)
458 Classical Electrodynamics
Or, using (13.107) for <0 2 >,
<0 2 > ~ 47tN[^^) In (2lOZ 1A )t (13.111)
\ pv I
The mean square angle increases linearly with the thickness t. But for
reasonable thicknesses such that the particle does not lose appreciable
energy, the Gaussian will still be peaked at very small forward angles.
The multiplescattering distribution for the projected angle of scattering
is
J.
V^T"^ <© 2 >
where both positive and negative values of d' are considered. The small
angle Rutherford formula (13.92) can be expressed in terms of the pro
JeCt6dangleaS *(«tfL (1 ,U3)
I dd' 2 \ pv / d' 3
This gives a singlescattering distribution for the projected angle :
P s (6') dd' = Nt^dd'=^ Nt{ 2 ^^ (13.114)
SV ' dd' 2 \ pv J d' z
The singlescattering distribution is valid only for angles large compared
to (& 2 Y A , and contributes a tail to the Gaussian distribution.
If we express angles in terms of the relative projected angle,
a = — ^ (13.115)
<0 2 )^ V
the multiple and singlescattering distributions can be written
1 _ 2
P m (ol) da. = —= e a doc
V 77 "
_ , N , 1 doc
P 8 (ol) da = ni ,^^_ v — z
(13.116)
81n(210Z^) a 6
where (13.1 1 1) has been used for (0 2 >. We note that the relative amounts
of multiple and single scatterings are independent of thickness in these
units, and depend only on Z. Even this Z dependence is not marked. The
factor 8 In (210Z _1 ^) has the value 36.0 for Z = 13 (aluminum) and the
value 31 .0 for Z = 82 (lead). Figure 13.8 shows the general behavior of the
scattering distributions as a function of a. The transition from multiple
to single scattering occurs in the neighborhood of a ^ 2.5. At this point the
Gaussian has a value of 1/600 times its peak value. Thus the singlescatter
ing distribution gives only a very small tail on the multiplescattering curve.
There are two things which cause departures from the simple behavior
shown in Fig. 13.8. The Gaussian shape is the limiting form of the
[Sect. 13.8]
Collisions between Charged Particles
459
Fig. 13.8 Multiple and single scattering distributions of projected angle. In the region
of plural scattering (a < 23) the dotted curve indicates the smooth transition from the
smallangle multiple scattering (approximately Gaussian in shape) to the wideangle
single scattering (proportional to a 3 ).
angular distribution for very large n. If the thickness / is such that n
(13.109) is not very large (i.e., n < 100), the distribution follows the single
scattering curve to smaller angles than a ^ 2.5, and is somewhat more
sharply peaked at zero angle than a Gaussian. On the other hand, if the
thickness is great enough, the mean square angle <0 2 > becomes comparable
with the angle max (13.102) which limits the angular width of the single
scattering distribution. For greater thicknesses the multiple scattering
curve extends in angle beyond the singlescattering region, so that there is
no singlescattering tail on the distribution (see Problem 13.5).
13.8 Electrical Conductivity of a Plasma
The considerations of multiple scattering can be applied rather directly
to the seemingly different problem of the electrical conductivity of a
460 Classical Electrodynamics
plasma. For simplicity we will consider the socalled Lorentz gas, which
consists of TV fixed ions of charge Ze per unit volume and NZ free electrons
per unit volume. Furthermore electronelectron interactions will be
ignored. The approximation of fixed ions is a reasonable one, at least for
plasmas with electrons and ions at roughly the same kinetic temperatures.
The effects of electronelectron collisions will be mentioned later.
The simple Drude theory of electrical conductivity, described briefly in
Section 7.8, is based on the single electron equation,
dv
m — = eE — mv\ (13.1 17)
dt
where v is the collision frequency. The lowfrequency electrical con
ductivity a due to electron motion is
NZe 2
a = =^ (13.118)
mv
The problem of calculating the proper collision frequency can be ap
proached by noting that the term mv\ in (13.1 17) really represents the rate
of decrease of forward momentum because of Coulomb collisions with the
ions as the electron moves under the action of the applied electric field. If
the scattering angle in a single elastic collision is 6, as indicated in Fig. 13.9,
the forward momentum lost by a particle of momentum/? is p(l — cos 6).
The average value of this quantity multiplied by the number of collisions
per unit distance is the loss in forward momentum per unit distance,
namely, mv. Thus
mv = Nap (1  cos d) (13.119)
where a here is the total cross section (13.104). Since all the Coulomb
scattering is at very small angles, <1 — cos 6) ~ \ <0 2 ). Then the forward
momentum loss per unit distance is
2\2
(Ze*>
mv ~ hNopiQ 2 ) = AttN^ In ^ (13.120)
mv
("max I
B m \J
Equation (13.106) has been used for (6 2 ). When (13.120) is inserted in
(13.118), we obtain a conductivity,
3
o(v)~ — — (13.121)
47T(Ze 2 ) In (0 ma x/0min)
This result holds for electrons of velocity v.
We now want to average over a thermal distribution. The variation
with v in (13.121) comes mainly from the factor t' 3 . The argument of the
logarithm can be evaluated at the mean velocity without introducing
appreciable error. At energies appropriate to even the hottest plasmas
nuclearsize effects are negligible. Consequently we put max = 1. The
value of 0min requires some discussion. For the screened atomic potential
[Sect. 13.8] Collisions between Charged Particles 461
P
Fig. 13.9
the result (13.97) or (13.98) was appropriate, with the atomic radius a
given by (13.95). For electronion collisions in a plasma the interaction
is the DebyeHuckel screened potential (10. 113). Consequently the Debye
length kj) 1 plays the role of the atomic radius a in the formulas for mln .
Either (13.97) or (13.98) is used, depending on which is larger. With these
substitutions the argument of the logarithm in (13.121) can be written
A = ^
12ttJV
" D (13.122)
Ze 2 \2itN
where k D is given by (10.106) or (10.112), and {u 2 Y A = kTjm. The upper
(lower) value of A is to be used when the mean electron energy f kT is less
(greater) than 13.6Z 2 electron volts.
The average value of the «th power of the magnitude of velocity for a
Maxwellian distribution is , N
/2JC7T* I 2 ; (m23)
\ m / T(f )
Consequently the value of the conductivity (13.121), averaged over a
Maxwellian velocity distribution, is
m (2 kT\ A ..., ....
<J— — „ (13.124)
Ze 2 In A W m J
This approximate result, obtained in a rather simpleminded way with the
elementary Drude theory, is within a factor of 2 of the correct value found
from an application of the Boltzmann equation. The physical mechanism
is the same in both calculations, but the more rigorous treatment involves
an averaging over v 5 rather than v z * Thus the two results differ by a
factor (v 5 )l(v 2 )(v 3 ) = 2.
* The added power of v 2 can be understood as follows. In the presence of the electric
field the formerly spherically symmetric velocity distribution tends to become distorted
in velocity directions parallel to the field. The amount of distortion determines the
current and, through Ohm's law, the conductivity. The distorted distribution results
from a balancing of the anisotropic electric force and the tendency towards isotropy
produced by the collisions. Since the scattering cross section varies as v~ 2 , the aniso
tropic part of the distribution has more highvelocity components than normal by a
factor y 2 .
462 Classical Electrodynamics
When electronelectron collisions are included, the forwardmomentum
loss is increased and so the conductivity is decreased from its value of
twice (13.124). The relative decrease depends on Z roughly as Z/(l + Z),
ranging from 0.58 for Z = 1 to 1.0 for Z> oo. Consequently (13.124)
as it stands can be used as a good approximation to the conductivity for a
hydrogen or deuterium plasma, including effects of electronelectron
collisions. If the classical (lowenergy) value of A (13.122) is used, (13.124)
can be written in the instructive form :
\()"A<», 3125 )
3 \77/
3 W/ In A
Since A is of the order of 10 4 for a typical hydrogen plasma (n e ~ 10 15
cm 3 , T~ 10 5 °K), a is ^200^ ^4x 10 14 sec" 1 . This is not quite as
large as metallic conductivities (~10 16 sec 1 ), but is sufficiently large that
the infinite conductivity approximation used in Chapter 10 is quite
adequate in discussing the penetration of fields into a plasma.
REFERENCES AND SUGGESTED READING
The problems of the penetration of particles through matter have interested Niels
Bohr all his life. A lovely presentation of the whole subject, with characteristic emphasis
on the interplay of classical and quantummechanical effects, appears in his compre
hensive review article of 1948:
Bohr.
Numerical tables and graphs of energyloss data, as well as key formulas, are given by
Rossi, Chapter 2,
Segre, article by H. A. Bethe and J. Ashkin.
Rossi also gives a semiclassical treatment of energy loss and scattering similar to ours.
The density effect on the energy loss by extremely relativistic particles is discussed,
with numerous results for different substances in graphical form, by
R. M. Sternheimer, Phys. Rev., 88, 851 (1952); 91, 256 (1953).
The correct calculation of the conductivity of a plasma is outlined by
Spitzer, Chapter 5.
PROBLEMS
13.1 A heavy particle of charge ze, mass M, and nonrelativistic velocity v
collides with a free electron of charge e and mass m initially at rest. With
no approximations, other than that of nonrelativistic motion and M > m,
show that the energy transferred to the electron in this Coulomb collision,
as a function of the impact parameter b, is
2(ze 2 ) 2 1
nw 2 b 2 + {ze 2 jmv 2 ) 2
[Probs. 13] Collisions between Charged Particles 463
13.2 {a) Taking/j<o>> = 12Zevin the quantummechanical energyloss formula,
calculate the rate of energy loss (in Mev/cm) in air at NTP, aluminum,
copper, lead for a proton and a mu meson, each with kinetic energies of
10, 100, 1000 Mev.
(b) Convert your results to energy loss in units of Mevcm 2 /gm and
compare the values obtained in different materials. Explain why all the
energy losses in Mevcm 2 /gm are within a factor of 2 of each other, whereas
the values in Mev/cm differ greatly.
13.3 Consider the energy loss by close collisions of a fast, but nonrelativistic,
heavy particle of charge ze passing through an electronic plasma. Assume
that the screened Coulomb interaction (10.113) acts between the electrons
and the incident particle.
(a) Show that the energy transfer in a collision at impact parameter b is
given approximately by
AE(b) ^^^ k^K^knb)
where m is the electron mass, v is the velocity of the incident particle, and
Icd is the Debye wave number (10.112).
(b) Determine the energy loss per unit distance traveled for collisions
with impact parameter greater than b min . Assuming kDb min < 1, write
down your result with both the classical and quantummechanical values
of b min .
13.4 With the same approximations as were used to discuss multiple scattering,
show that the projected transverse displacement y (see Fig. 13.7) of an
incident particle is described approximately by a Gaussian distribution,
P(y) dy = A exp
y*
2<2/ 2 >.
dy
where the mean square displacement is (y 2 ) = (x 2 /6)<0 2 >, x being the
thickness of the material traversed and <© 2 > the mean square angle of
scattering.
13.5 If the finite size of the nucleus is taken into account in the "singlescattering"
tail of the multiplescattering distribution, there is a critical thickness x c
beyond which the singlescattering tail is absent.
(a) Define x e in a reasonable way and calculate its value (in cm) for
aluminum and lead, assuming that the incident particle is relativistic.
(b) For these thicknesses calculate the number of collisions which occur
and determine whether the Gaussian approximation is valid.
14
Radiation by Moving Charges
It is well known that accelerated charges emit electromagnetic
radiation. In Chapter 9 we discussed examples of radiation by macroscopic
timevarying charge and current densities, which are fundamentally charges
in motion. We will return to such problems in Chapter 16 where multipole
radiation is treated in a systematic way. But there is a class of radiation
phenomena where the source is a moving point charge or a small number
of such charges. In these problems it is useful to develop the formalism in
such a way that the radiation intensity and polarization are related
directly to properties of the charge's trajectory and motion. Of particular
interest are the total radiation emitted, the angular distribution of radiation,
and its frequency spectrum. For nonrelativistic motion the radiation is
described by the wellknown Larmor result (see Section 14.2). But for
relativistic particles a number of unusual and interesting effects appear.
It is these relativistic aspects which we wish to emphasize. In the present
chapter a number of general results are derived and applied to examples of
charges undergoing prescribed motions, especially in external force fields.
Chapter 15 deals with radiation emitted in atomic or nuclear collisions.
14.1 LienardWiechert Potentials and Fields for a Point Charge
In Chapter 6 it was shown that for localized charge and current distri
butions without boundary surfaces the scalar and vector potentials can be
written as
AJx, t) =  j\ Jtl(X ' ,t) d ( t ' + ~  ') #*' dt' (14.1)
where R = (x — x'), and the delta function provides the retarded behavior
464
[Sect. 14.1] Radiation by Moving Charges 465
demanded by causality. For a point charge e with velocity c(S(?) at the
point r(7) the chargecurrent density is
J Ax, = ec^ d[x  r(0] (14.2)
where ^ = (p, /)• With tnis source density the spatial integration in (14.1)
can be done immediately, yielding
A ax, t) = e\ — — o
f' +
R(t')
— t
dt'
(14.3)
where now R(t') = \x — r(t')\. Although (14.3) is a convenient form to
utilize in calculating the fields, the integral over dt' can be performed,
provided we recall from Section 1.2 that when the argument of the delta
function is a function of the variable of integration the standard results are
modified as follows :
/■
g(*) d[f( x ) ci]dx =
g( x )
UfldxJ
The function f(t') = t' + [R(t')/c] has a derivative
f(x) = a
df 1 , ldR , a
(14.4)
(14.5)
dt' c dt'
where cp is the instantaneous velocity of the particle, and n = R/i? is a
unit vector directed from the position of the charge to the observation
point. With (14.5) in (14.4) and (14.3) the potentials of the point charge,
called the Lienard Wiechert potentials, are
0(x, t) = e
A(x, t) = e
J_
A
kR.
(14.6)
The square bracket with subscript ret means that the quantity in brackets
is to be evaluated at the retarded time, t' = t — [R(t)/c]. We note that,
for nonrelativistic motion, k » 1. Then the potentials (14.6) reduce to the
wellknown nonrelativistic results.
To determine the fields E and B from the potentials A^ it is possible to
perform the specified differential operations directly on (14.6). But this is
a more tedious procedure than working with the form (14.3). We note that
in (14.3) the only dependence on the spatial coordinates x of the observa
tion point is through R. Hence the gradient operation is equivalent to
d d
MR
dR
= n
dR
(14.7)
466 Classical Electrodynamics
Consequently the electric and magnetic fields can be written as
d
dt'
(14.8)
f X + ^f)
B(x,r) = . (nx(3) — £  + J«5'(V+*A
The primes on the delta functions mean differentiation with respect to
their arguments. If the variable of integration is changed to/(r') = t' +
[R(t')/c], we can integrate by parts on the derivative of the delta function.
Then we find readily
E(x, = e
B(x, t) = e
+ J_ d_ (n  p
ret
.kR 2 ck dt' \ kR
> x n [ 1 d / ft x n \"
. kR 2 CKdt' \ kR /_
ret
(14.9)
It is convenient to perform first the differentiation of the unit vector n.
It is evident from Fig. 14.1 that the rate of change of n with time is the
negative of the ratio of the perpendicular component of v to R. Thus
_rfn _ n x (n x P)
cdt' R
(14.10)
When we perform the differentiation of n wherever it appears explicitly,
we obtain
E(x, t) = e
B(x, = e
"_E_ + H_± l±\ _ JL _ 1 A (IX
K 2 R* CKdt'\KRJ K 2 R 2 CKdt'\KR!.
\k 2 R 2 CKdt'\KR/J
(14.11)
We observe at this point that the magnetic induction is related simply to
Mt')
Fig. 14.1
[Sect. 14.1]
Radiation by Moving Charges
467
the electric field by the relation,
B = n x E (14.12)
where the equation is understood to be in terms of the retarded quantities
in square brackets.
The remaining derivatives needed in (14.11) are
dt'
cdt' c
Then the electric field can be written
E(x, = e
(n  pXli 2 )"
ret c
f x {(n  0) x p}
Lk 3 R
_ ret
(14.13)
(14.14)
while the magnetic induction is given by (14.12). Fields (14.12) and (14.14)
divide themselves naturally into "velocity fields," which are independent
of acceleration, and "acceleration fields," which depend linearly on p.
The velocity fields are essentially static fields falling off as Rr 2 , whereas
the acceleration fields are typical radiation fields, both E and B being
transverse to the radius vector and varying as U 1 .
For a particle in uniform motion the velocity fields must be the same as
those obtained in Section 11.10 by means of a Lorentz transformation on
the static Coulomb field. For example, the transverse electric field E x at a
point a perpendicular distance b from the straight line path of the charge
was found to be
EJt) = ey o3 , (14.15)
The origin of the time t is chosen so that the charge is closest to the
observation point at t = 0. The electric field E t (t) given by (14.15) bears
little resemblance to the velocity field in (14.14). The reason for this
apparent difference is that field (14.15) is expressed in terms of the present
position of the charge rather than its retarded position. To show the
equivalence of the two expressions we consider the geometrical configura
tion shown in Fig. 14.2. Here O is the observation point, and the points P
and P' are the present and apparent or retarded positions of the charge at
time t. The distance P'Q is pR cos = (n • p)fl. Therefore the distance
OQ is kR. But from triangles OPQ and PP'Q we find
( K Rf = r 2  (PQ) 2 = r 2  p*(R sin 0) 2
Then from triangle OMP' we have R sin 6 = b, so that
(kR) 2 =b 2 + vH 2  A 2 = " 2 (b 2 + A 2 * 2 ) (14.16)
468
Classical Electrodynamics
Fig. 14.2 Present and retarded positions of a charge in uniform motion.
The transverse component of the velocity field in (14.14) is
b
£i(0 = e
.y\ K RfS
ret
(14.17)
With substitution (14.16) for kR in terms of the charge's present position,
we find that (14.17) is equal to (14.15). The other components of E and B
come out similarly.
14.2 Total Power Radiated by an Accelerated Charge — Larmor's
Formula and Its Relativistic Generalization
If a charge is accelerated but is observed in a reference frame where its
velocity is small compared to that of light, then in that coordinate frame
the acceleration field in (14.14) reduces to
n x (n x (J)
E„ = e 
C L
R
The instantaneous energy flux is given by the Poynting's vector,
S = ExB= EJ 2 n
47T 4"7T
This means that the power radiated per unit solid angle is*
§■ = y ItfEJ 2 =/i«x(nx p) 2
ail 47r 4nc
* In writing angular distributions of radiation we will always exhibit the polariza
tion explicitly by writing the absolute square of a vector which is proportional to the
electric field.
(14.18)
(14.19)
(14.20)
[Sect. 14.2]
Radiation by Moving Charges
469
If is the angle between the acceleration v and n, as shown in Fig. 14.3,
then the power radiated can be written
dP
dQ.
= J—i? sin 2
47TC 3
(14.21)
This exhibits the characteristic sin 2 angular dependence which is a well
known result. We note from (14.18) that the radiation is polarized in the
plane containing v and n. The total instantaneous power radiated is
obtained by integrating (14.21) over all solid angle. Thus
3 c 3
(14.22)
This is the familiar Larmor result for a nonrelativistic, accelerated charge.
Larmor's formula (14.22) can be generalized by arguments about
covariance under Lorentz transformations to yield a result which is valid
for arbitrary velocities of the charge. Radiated electromagnetic energy
behaves under Lorentz transformation like the fourth component of a
4vector (see Problem 11.13). Since dE mA = P dt, this means that the
power P is a Lorentz invariant quantity. If we can find a Lorentz invariant
which reduces to the Larmor formula (14.22) for £ < 1, then we have the
desired generalization. There are, of course, many Lorentz invariants
which reduce to the desired form when £>0. But from (14.14) it is
evident that the general result must involve only p and p. With this
restriction on the order of derivatives which can appear the result is unique.
To find the appropriate generalization we write Larmor's formula in the
suggestive form :
" 2 f^E.^) (14.23)
\dt dt!
P = 2 
Fig. 14.3
470 Classical Electrodynamics
where m is the mass of the charge, and p its momentum. The Lorentz
invariant generalization is clearly
P = ? 4 3 l^) (14.24)
3 ra V V dr dr!
where dr = dtjy is the proper time element, and/?^ is the charged particle's
momentumenergy 4 vector.* To check that (14.24) reduces properly to
(14.23) as /? ■> we evaluate the 4vector scalar product,
dp,dp IL= (dpf_i (dE) 2 = (dpY _ * (dpY (u 2$)
dr dr \drf c 2 \dr I W H \dr!
If (14.24) is expressed in terms of the velocity and acceleration by means
of E = ymc 2 and p = ytm, we obtain the Lienard result (1898):
i } = ;7W(P><M (14.26)
3 c
One area of application of the relativistic expression for radiated power
is that of chargedparticle accelerators. Radiation losses are sometimes
the limiting factor in the maximum practical energy attainable. For a
given applied force (i.e., a given rate of change of momentum) the radiated
power (14.24) depends inversely on the square of the mass of the particle
involved. Consequently these radiative effects are largest for electrons.
We will restrict our discussion to them.
In a linear accelerator the motion is one dimensional. From (14.25) it
is evident that in that case the radiated power is
P = \4 iff (14.27)
3 m 2 c 3 \dt I
The rate of change of momentum is equal to the change in energy of the
particle per unit distance. Consequently
P== ?jM^) 2 (14.28)
3 mV \dx) K
showing that for linear motion the power radiated depends only on the
external forces which determine the rate of change of particle energy with
distance, not on the actual energy or momentum of the particle. The ratio
* That (14.24) is unique can be seen by noting that a Lorentz invariant is formed by
taking scalar products of 4vectors or higherrank tensors. The available 4vectors are
Pp and dp^/ch. Only form (14.24) reduces to the Larmor formula for /S »■ 0. Contraction
of higherrank tensors such as p^idpjdr) can be shown to vanish, or to give results
proportional to (14.24) or m 2 .
[Sect. 14.2] Radiation by Moving Charges 471
of power radiated to power supplied by the external sources is
__P_ = 2 J_ 1 dE _ 2 (e 2 lmc 2 ) dE (J4 29)
(d£/dO ~~ 3 m 2 c 3 i> d* 3 mc 2 dx
where the last form holds for relativistic particles (/5>l). Equation (14.29)
shows that the radiation loss will be unimportant unless the gain in
energy is of the order of mc 2 = 0.511 Mev in a distance of e>c 2 =
2.82 x 10 13 cm, or of the order of 2 X 10 14 Mev/meter! Typical energy
gains are less than 10 Mev/meter. Radiation losses are completely
negligible in linear accelerators.
Circumstances change drastically in circular accelerators like the
synchrotron or betatron. In such machines the momentum p changes
rapidly in direction as the particle rotates, but the change in energy per
revolution is small. This means that
dp
dr
= yco\v\>^ (1430)
c dr
Then the radiated power (14.24) can be written approximately
p=?4i^ 2 ipi 2 =? £ T^ 4 (1431)
3 mV 3 p*
where we have used co = (cft/p), p being the orbit radius. This result was
first obtained by Lienard in 1898. The radiativeenergy loss per revolution
is
dE = llL£p = ±L e lpy (14.32)
cP 3 P
For highenergy electrons (/? ^ 1) this has the numerical value,
dE (Mev) = 8.85 X 10" 2 [£(Bev)] ' (14.33)
p (meters)
For a typical lowenergy synchroton, p ^ 1 meter, .E max ^0.3 Bev.
Hence, (5£ max ~ 1 Kev per revolution. This is less than, but not negligible
compared to, the energy gain of a few kilovolts per turn. In the largest
electron synchrotrons, the orbit radius is of the order of 10 meters and the
maximum energy is 5 Bev. Then the radiative loss is ~5.5 Mev per revolu
tion. Since it is extremely difficult to generate radiofrequency power at
levels high enough to produce energy increments much greater than this
amount per revolution, it appears that 510 Bev is an upper limit on the
maximum energy of circular electron accelerators.
472 Classical Electrodynamics
The power radiated in circular accelerators can be expressed numerically
as
_, „. 10 6 <5£(Mev) _, . ,<aia\
P (watts) = J (amp) (14.34)
2tt p (meters)
where / is the circulating beam current. This equation is valid if the
emission of radiation from the different electrons in the circulating beam
is incoherent. In the largest electron synchrotrons the radiated power
amounts to 0.1 watt per microampere of beam. Although this power
dissipation is very small the radiated energy is readily detected and has
some interesting properties which will be discussed in Section 14.6.
14.3 Angular Distribution of Radiation Emitted by an
Accelerated Charge
For an accelerated charge in nonrelativistic motion the angular distri
bution shows a simple sin 2 behavior, as given by (14.21), where is
measured relative to the direction of acceleration. For relativistic motion
the acceleration fields depend on the velocity as well as the acceleration.
Consequently the angular distribution is more complicated. From (14.14)
the radial component of Poynting's vector can be calculated to be
e 2
[S • n]ret =
Aire
1 n x [(n  p) x p]
U«R 2
ret
(14.35)
It is evident that there are two types of relativistic effect present. One is
the effect of the specific spatial relationship between (3 and (3, which will
determine the detailed angular distribution. The other is a general, relati
vistic effect arising from the transformation from the rest frame of the
particle to the observer's frame and manifesting itself by the presence of
the factors k (14.5) in the denominator of (14.35). For ultrarelativistic
particles the latter effect dominates the whole angular distribution.
In (14.35) S • n is the energy per unit area per unit time detected at an
observation point at time t due to radiation emitted by the charge at time
t' = t — R{t')jc, If we wanted to calculate the energy radiated during a
finite period of acceleration, say from /' = T x to t' = T 2 , we would write
W=\ [Sn] ret ^= (Sn) ^dt' (14.36)
Thus we see that the useful and meaningful quantity is (S • n) (dt/dt'), the
power radiated per unit area in terms of the charge's own time. We
[Sect. 14.3]
Radiation by Moving Charges
473
Fig. 14.4 Radiation pattern for
charge accelerated in its direction
of motion. The two patterns are
not to scale, the relativistic one
(appropriate for y — 2) having
been reduced by a factor ~10 2
for the same acceleration.
therefore define the power radiated per unit solid angle to be
dP(n = R\S.n)— = KR 2 S'n
dQ.
dt
dt'
(14.37)
If we imagine the charge to be accelerated only for a short time during
which 3 and (3 are essentially constant in direction and magnitude, and we
observe the radiation far enough away from the charge that n and R
change negligibly during the acceleration interval, then (14.37) is pro
portional to the angular distribution of the energy radiated. With (14.35)
for the Poynting's vector, the angular distribution is
dP(t') _ e* In x {(n  ft) x (3} 2
da
Attc (1  n • p) 5
(14.38)
The simplest example of (14.38) is linear motion in which (3 and (3 are
parallel. If 6 is the angle of observation measured from the common
direction of (3 and (3, then (14.38) reduces to
dP(t')
dQ
e v
sin
4ttc 3 (1  cos Of
(14.39)
For /? < 1, this is the Larmor result (14.21). But as /S > 1, the angular
distribution is tipped forward more and more and increases in magnitude,
as indicated schematically in Fig. 14.4. The angle max for which the
intensity is a maximum is
= cos
J(Vl + 15/Pl)
L3p
2y
(14.40)
where the last form is the limiting value for /S — >• 1 . In this same limit the
peak intensity is proportional to y 8 . Even for /3 = 0.5, corresponding to
electrons of <~80 Kev kinetic energy, m ax = 38.2°. For relativistic
particles, m ax is very small, being of the order of the ratio of the rest
energy of the particle to its total energy. Thus the angular distribution is
474
Classical Electrodynamics
confined to a very narrow cone in the direction of motion. For such small
angles the angular distribution (14.39) can be written approximately
dQ. 77
(ydf
(i + y 2 ey
(14.41)
The natural angular unit is evidently y _1 . The angular distribution is
shown in Fig. 14.5 with angles measured in these units. The peak occurs
at yd = , and the halfpower points at yd = 0.23 and yd = 0.91. The
root mean square angle of emission of radiation in the relativistic limit is
(d 2 ) 1A = A = —
y E
(14.42)
This is typical of the relativistic radiation patterns, regardless of the
vectorial relation between (3 and (3. The total power radiated can be
obtained by integrating (14.39) over all angles. Thus
2 p 2
P(t') = "vY
3 c
(14.43)
in agreement with (14.26) and (14.27).
Another example of angular distribution of radiation is that for a charge
in instantaneously circular motion with its acceleration (3 perpendicular
to its velocity (3. We choose a coordinate system such that instantaneously
(3 is in the z direction and (3 is in the x direction. With the customary polar
angles 6, <f> defining the direction of observation, as shown in Fig. 14.6, the
general formula (14.38) reduces to
dP(t')
dQ.
1
4ttc 3 (1/9 cos Of
1 
sin 2 6 cos 2 <£
, 2 (1
P cos Of A
(14.44)
We note that, although the detailed angular distribution is different from
the linear acceleration case, the same characteristic relativistic peaking at
forward angles is present. In the relativistic limit (y > 1), the angular
Fig. 14.5 Angular distribution of
radiation for relativistic particle.
[Sect. 14.4]
Radiation by Moving Charges
475
distribution can be written approximately
dP(t')
l e JL
77 C 3
1
1 
4y 2 2 cos 2 <f
(1 + r 202 )2 j
(14.45)
dQ. 77 c 3 ' (1 + y 2 2 ) 3
The root mean square angle of emission in this approximation is given by
(14.42), just as for onedimensional motion. The total power radiated can
be found by integrating (14.44) over all angles or from (14.26):
«0 = lfr>
(14.46)
It is instructive to compare the power radiated for acceleration parallel
to the velocity (14.43) or (14.27) with the power radiated for acceleration
perpendicular to the velocity (14.46) for the same magnitude of applied
force. For circular motion, the magnitude of the rate of change of
momentum (which is equal to the applied force) is ymv. Consequently,
(14.46) can be written 2
■» circular \* ) —
3 mV
r
©•
(14.47)
When this is compared to the corresponding result (14.27) for rectilinear
motion, we find that for a given magnitude of applied force the radiation
emitted with a transverse acceleration is a factor of y % larger than with a
parallel acceleration.
14.4 Radiation Emitted by a Charge in Arbitrary, Extreme Relativistic
Motion
For a charged particle undergoing arbitrary, extreme relativistic motion
the radiation emitted at any instant can be thought of as a coherent super
position of contributions coming from the components of acceleration
476 Classical Electrodynamics
parallel to and perpendicular to the velocity. But we have just seen that for
comparable parallel and perpendicular forces the radiation from the parallel
component is negligible (of order 1/y 2 ) compared to that from the perpen
dicular component. Consequently we may neglect the parallel component
of acceleration and approximate the radiation intensity by that due to the
perpendicular component alone. In other words, the radiation emitted by
a charged particle in arbitrary, extreme relativistic motion is approxi
mately the same as that emitted by a particle moving instantaneously along
the arc of a circular path whose radius of curvature p is given by
v 2 c 2
P = — ~— (14.48)
where v x is the perpendicular component of acceleration. The form of the
angular distribution of radiation is (14.44) or (14.45). It corresponds to a
narrow cone or searchlight beam of radiation directed along the instanta
neous velocity vector of the charge.
For an observer with a frequencysensitive detector the confinement of
the radiation to a narrow pencil parallel to the velocity has important
consequences. The radiation will be visible only when the particle's
velocity is directed towards the observer. For a particle in arbitrary
motion the observer will detect a pulse or burst of radiation of very short
time duration (or a succession of such bursts if the particle is in periodic
motion), as sketched in Fig. 14.7. Since the angular width of the beam is of
the order of y 1 , the particle will illuminate the observer only for a time
interval
cy
in terms of its own time, where p is the radius of curvature (14.48). The
observer sees, however, a time interval,
\dt'/
where (dt/dt') = (k) ~ (1/y 2 ). Consequently the duration of the burst of
radiation at the detector is
Af ~ i £ (14.49)
y 6 c
A pulse of this duration will contain, according to general arguments about
Fourier integrals (see Section 7.3), appreciable frequency components up to
a critical frequency, co c , of the order of
^~()y 3 04.50)
For circular motion cjp is the angular frequency of rotation co , and even
[Sect. 14.5]
Radiation by Moving Charges
All
— — — O
Pit)
Koo)
Fig. 14.7 Radiating particle illuminates the detector at O only for a time A/. The
frequency spectrum thus contains frequencies up to a maximum io c ~ (Af) _1 .
for arbitrary motion it plays the role of a fundamental frequency of motion.
Equation (14.50) shows that a relativistic particle emits a broad spectrum
of frequencies if E > mc 2 , up to y 3 times the fundamental frequency. In
a 200Mev synchrotron, y m ax ^ 400. Therefore co c ~ 6 x 10 7 eo . Since
the rotation frequency is a> ~ 3 X 10 8 sec 1 , the frequency spectrum
of emitted radiation extends up to ~2 x 10 16 sec 1 . This represents a
wavelength of 1000 angstroms. Hence the spectrum extends beyond the
visible, even though the fundamental frequency is in the 100Mc range.
In Section 14.6 we will discuss in detail the angular distribution of the
different frequency components, as well as the total energy radiated as a
function of frequency.
14.5 Distribution in Frequency and Angle of Energy Radiated by
Accelerated Charges
The qualitative arguments of the previous section show that for relati
vistic motion the radiated energy is spread over a wide range of frequencies.
478 Classical Electrodynamics
The range of the frequency spectrum was estimated by appealing to
properties of Fourier integrals. The argument can be made precise and
quantitative by the use of Parseval's theorem of Fourier analysis.
The general form of the power radiated per unit solid angle is
^=A«P (.4.51)
where
A(0 = (^j 2 [KE] re t (14.52)
E being the electric field (14.14). In (14.51) the instantaneous power is
expressed in the observer's time (contrary to the definition in Section 14.3),
since we wish to consider a frequency spectrum in terms of the observer's
frequencies. For definiteness we think of the acceleration occurring for
some finite interval of time, or at least falling off for remote past and
future times, so that the total energy radiated is finite. Furthermore, the
observation point is considered far enough away from the charge that the
spatial region spanned by the charge while accelerated subtends a small
solid angle element at the observation point.
The total energy radiated per unit solid angle is the time integral of
(14.51):
^=J_JA(0l 2 ^ (14.53)
This can be expressed alternatively as an integral over a frequency
spectrum by use of Fourier transforms. We introduce the Fourier transform
A(co) of A(t),
A(eo) = L A(t)e i0it dt (14.54)
•v Z7T J co
and its inverse,
Then (14.53) can be written
A(0 = L J A(w)e~ ilot dm (14.55)
a Z7T J oo
dw i r°° c° f°°
—  = — dt\ dco] dco' A*(o>') • A{(o)e i{( °' m)t (14.56)
dil Z7T J co J oo J co
Interchanging the orders of time and frequency integration, we see that the
time integral is just a Fourier representation of the delta function
d((o' — o>). Consequently the energy radiated per unit solid angle becomes
4? = I \Mco)\ 2 dco (14.57)
d\l J co
dQ Jo
[Sect. 14.5] Radiation by Moving Charges 479
The equality of (14.57) and (14.53), with suitable mathematical restrictions
on the function A(0, is a special case of Parseval's theorem. It is customary
to integrate only over positive frequencies, since the sign of the frequency
has no physical meaning. Then the relation,
d ^da> (14.58)
dQ.
defines a quantity dI{(o)jdQ which is the energy radiated per unit solid
angle per unit frequency interval :
^ = A(co) 2 + A(co) 2 (14.59)
dQ.
If A(0 is real, from (14.55) it is evident that A(co) = A*(o>). Then
^^ = 2 A(o>) 2 (14.60)
dQ
This result relates in a quantitative way the behavior of the power radiated
as a function of time to the frequency spectrum of the energy radiated.
By using (14.14) for the electric field of an accelerated charge we can
obtain a general expression for the energy radiated per unit solid angle
per unit frequency interval in terms of an integral over the trajectory of
the particle. We must calculate the Fourier transform (14.54) of A(0 given
by (14.52). Using (14.14), we find
A^ ( * V f W ,;4 nx[(n(3)x $]"
where ret means evaluated at t' + [R(t')/c] = t. We change the variable
of integration from t to t', thereby obtaining the result:
Ma) = (4.)* r  ^ + i«™ ■»[(— p*M „, (14 . 62)
\87rV Joc KT
Since the observation point is assumed to be far away from the region of
space where the acceleration occurs, the unit vector n is sensibly constant
in time. Furthermore the distance R(t') can be approximated as
R(t') ~ x  n • r(0 (14.63)
where x is the distance from an origin O to the observation point P, and
r(r') is the position of the particle relative to O, as shown in Fig. 14.8. Then,
apart from an overall phase factor, (14.62) becomes
A((o) = (AT* f 00 ^i[..ko/.]) nx[(nP)xM df (14 64)
WV J 00 K 2
The primes on the time variable have been omitted for brevity. The
dt (14.61)
_ ret
480
Classical Electrodynamics
R(t')
Fig. 14.8
energy radiated per unit solid angle per unit frequency interval (14.60) is
accordingly
dl(a>)
d£l
4tt 2 c
f * n x [(n  (3) x
Joo (1SiO 2
P] i«>(t[nt(t)lc])
dt
(14.65)
For a specified motion r(t) is known, (3(?) and (3(0 can be computed, and the
integral can be evaluated as a function of co and the direction of n. If
accelerated motion of more than one charge is involved, a coherent sum
of amplitudes A/co), one for each charge, must replace the single amplitude
in (14.65) (see Problems 14.11, 15.2, and 15.3).
Even though (14.65) has the virtue that the time interval of integration
is explicitly shown to be confined to times for which the acceleration is
different from zero, a simpler expression for some purposes can be
obtained by an integration by parts in (14.64). It is easy to demonstrate
that the integrand in (14.64), excluding the exponential, is a perfect
differential :
(14.66)
n x [(n 
K 2
P) x M _ d
dt
n x (n x P)
K
ration by parts leads to the intensity distribut
dI(co) _ e 2 eo 2
dtl 4tt 2 c
n x (n x
J — oo
q\ i(o(t[nt(t)lc
]) dt
(14.67)
It should be observed that in (14.67) and (14.65) the polarization of the
emitted radiation is specified by the direction of the vector integral in each.
The intensity of radiation of a certain fixed polarization can be obtained
by taking the scalar product of the appropriate unit polarization vector
with the vector integral before forming the absolute square.
For a number of charges e i in accelerated motion the integrand in (14.67)
involves the replacement,
*\e$fi m ' l *' m (14.68)
3=1
efie
—i(co/c)nt(t)
[Sect. 14.6] Radiation by Moving Charges 481
In the limit of a continuous distribution of charge in motion the sum over
j becomes an integral over the current density J(x, t) :
e p e iMOnm _^ 1 t d s x J(x? f)e .•(«/«)«•« (14#69)
Then the intensity distribution becomes
^ = ^LUnx[nxJ(x > 0>
d£l 4wV \J J
ia>[t(nx)lc]
(14.70)
a result which can be obtained from the direct solution of the inhomo
geneous wave equation for the vector potential (14.1).
Of some interest is the radiation associated with a moving magnetic
moment. This can be most easily expressed by recalling from Chapter 5
that a magnetization density M(x., t) is equivalent to a current,
J M = C V x M (14.71)
Then substitution into (14.70) yields
dI M (co)_ a, 4 \j dt j^ xnxJ(iXtt)e ^in^
d£l 4tt 2 c
2„3
(14.72)
If the magnetization is a point magnetic moment (*,(?) at the point r(t),
th6n "(x, = tfO d[x  1(01 (1473)
and the energy radiated per unit solid angle per unit frequency interval is
f M («>) _ ft) 4 I f
dO. ~4tt 2 c 3 J
Anx^ 1 '*'
(14.74)
dl,
dO.
We note that there is a characteristic difference of a factor co 2 between the
radiated intensity from a magnetic dipole and an accelerated charge, apart
from the frequency dependence of the integrals.
The general formulas developed in this section, especially (14.65) and
(14.67), will be applied in this chapter and subsequent ones to various
problems involving the emission of radiation. The magneticmoment
formula (14.74) will be applied to the problem of radiation emitted in
orbitalelectron capture by nuclei in Chapter 15.
14.6 Frequency Spectrum of Radiation Emitted by a Relativistic Charged
Particle in Instantaneously Circular Motion
In Section 14.4 we saw that the radiation emitted by an extremely
relativistic particle subject to arbitrary accelerations is equivalent to that
482
Classical Electrodynamics
Fig. 14.9
emitted by a particle moving instantaneously at constant speed on an
appropriate circular path. The radiation is beamed in a narrow cone in
the direction of the velocity vector, and is seen by the observer as a short
pulse of radiation as the searchlight beam sweeps across the observation
point.
To find the distribution of energy in frequency and angle it is necessary
to calculate the integral in (14.67). Because the duration of the pulse
A?' ~ (p/cy) is very short, it is necessary to know the velocity (3 and
position r(t) over only a small arc of the trajectory whose tangent points
in the general direction of the observation point. Figure 14.9 shows an
appropriate coordinate system. The segment of trajectory lies in the xy
plane with instantaneous radius of curvature p. Since an integral will be
taken over the path, the unit vector n can be chosen without loss of
generality to lie in the xz plane, making an angle 6 (the colatitude) with
the x axis. Only for very small 6 will there be appreciable radiation
intensity. The origin of time is chosen so that at t = the particle is at the
origin of coordinates.
The vector part of the integrand in (14.67) can be written
n x (n x p) = £
€,, sin
() + e± cos () sin d (14.75)
where e„ = e 2 is a unit vector in the y direction, corresponding to polariza
tion in the plane of the orbit; c ± = n x e 2 is the orthogonal polariza
tion vector corresponding approximately to polarization perpendicular to
the orbit plane (for 6 small). The argument of the exponential is
CO
(■f>)[«H 2 )
cos
(14.76)
483
[Sect. 14.6] Radiation by Moving Charges
Since we are concerned with small angles and comparatively short times
around t = 0, we can expand both trigonometric functions in (14.76) to
obtain
{^Mfa ')'+&<
3
(14.77)
where (3 has been put equal to unity wherever possible. Using the time
estimate p\cy for / and the estimate (d 2 Y A (14.42) for 0, it is easy to see that
neglected terms in (14.77) are of the order of y~ 2 times those kept.
With the same type of approximations in (14.75) as led to (14.77), the
radiatedenergy distribution (14.67) can be written
dI(co)
dO. 4n 2 c
where the amplitudes are*
e H A H (m) + € 1 i4 1 (ft))
(14.78)
^»>= i6 i>('f[fe +9, ) ,+ Sl) < "
(14.79)
A change of variable to x =
parameter £,
r / 1 \ *i
ctjp\ h 2 l and introduction of the
£ = ^(l + 2 f (14.80)
allows us to transform the integrals in A n (co) and A L {w) into the form:
A n (<o) = "( 2 + 02 ) \ " ^^P V&( z + *^ dx
A x ((o) = ^ e(\ + 2 ) f °° exp [%g(z + i* 8 )] dx
(14.81)
* The fact that the limits of integration in (14.79) are t = ± oo may seem to contradict
the approximations made in going from (14.76) to (14.77). The point is that for most
frequencies the phase of the integrands in (14.79) oscillates very rapidly and makes the
integrands effectively zero for times much smaller than those necessary to maintain the
validity of (14.77). Hence the upper and lower limits on the integrals can be taken as
infinite without error. Only for frequencies of the order of a> ~ (pip) ~ eo„ does the
approximation fail. But we have seen in Section 14.4 that for relativistic particles essenti
ally all the frequency spectrum is at much higher frequencies.
484
Classical Electrodynamics
The integrals in (14.81) are identifiable as Airy integrals, or alternatively
as modified Bessel functions :
I
1
x sin [f (x + * x 3 )] dx = — K H (£)
o V3
i
1
cos [f i(x + z 3 )] <*# = — K H (£)
o 73
(14.82)
Consequently the energy radiated per unit frequency interval per unit
solid angle is
dljco)
dQ
3tt 2 c
m
+
Kk\S) +
d 2
(My 2 ) + &
^m
(14.83)
The first term in the square bracket corresponds to radiation polarized in
the plane of the orbit, and the second to radiation polarized perpendicular
to that plane.
We now proceed to examine this somewhat complex result. First we
integrate over all frequencies and find that the distribution of energy in
angle is
f
Jo
dI((o) , 1 e 2
dco =
1
dn
16 P
(i+*T
1 +
7(l/y 2 ) + 2 J
(14.84)
vy
This shows the characteristic behavior seen in Section 14.3. Equation
(14.84) can be obtained directly, of course, by integrating a slight generali
zation of the circularmotion power formula (14.44) over all times. As in
(14.83), the first term in (14.84) corresponds to polarization parallel to the
orbital plane, and the second to perpendicular polarization. Integrating
over all angles, we find that seven times as much energy is radiated with
parallel polarization as with perpendicular polarization. The radiation
from a relativistically moving charge is very strongly, but not completely,
polarized in the plane of motion.
The properties of the modified Bessel functions summarized in (3.103)
and (3.104) show that the intensity of radiation is negligible for £ > 1.
From (14.80) we see that this will occur at large angles; the greater the
frequency, the smaller the critical angle beyond which there will be
negligible radiation. This shows that the radiation is largely confined to
the plane containing the motion, as shown by (14.84), being more so
confined the higher the frequency relative to dp. If to gets too large,
however, we see that  will be large at all angles. Then there will be
negligible total energy emitted at that frequency. The critical frequency
[Sect. 14.6]
Radiation by Moving Charges
485
co c beyond which there is negligible radiation at any angle can be denned
by £ = 1 for = 0. Then we find
*(H(J e
(14.85)
This critical frequency is seen to agree with our qualitative estimate (14.50)
of Section 14.4. If the motion of the charge is truly circular, then cjp is
the fundamental frequency of rotation, co . Then we can define a critical
harmonic frequency to c = n c co , with harmonic number,
\m<rf
(14.86)
Since the radiation is predominantly in the orbital plane for y > 1, it is
instructive to evaluate the angular distribution (14.83) at = 0. For
frequencies well below the critical frequency (co < co c ), we find
dl{co)
da
e=o
?]W
For the opposite limit of w > co c , the result is
dI((o)
dQ
e=o
jL £_ v 2 — p 2o>/<»c
27T C
CO _.
co„
(14.87)
(14.88)
These limiting forms show that the spectrum at 6 = increases with
frequency roughly as co 2/3 well below the critical frequency, reaches a
maximum in the neighborhood of co c , and then drops exponentially to
zero above that frequency.
The spread in angle at a fixed frequency can be estimated by determining
the angle C at which £(0 C ) ~ 1(0) +1. In the lowfrequency range
(a> < co c ), £(0) is very small, so that £(0 C ) — 1 This g ives
\cop' y\co'
(14.89)
We note that the lowfrequency components are emitted at much wider
angles than the average, <0 2 )^ ~ y" 1 . In the highfrequency limit
(o> > co c ), 1(0) is large compared to unity. Then the intensity falls off in
angle approximately as
dI(co) dI(co)
3<oy 2 2 /(o e
dQ.
dQ.
(14.90)
9 =
486 Classical Electrodynamics
Thus the critical angle, defined by the \\e point, is
y \3o)/
(14.91)
This shows that the highfrequency components are confined to an angular
range much smaller than average. Figure 14.10 shows qualitatively the
angular distribution for frequencies small compared with, of the order of,
and much larger than, co c . The natural unit of angle yd is used.
The frequency distribution of the total energy emitted as the particle
passes by can be found by integrating (14.83) over angles:
I(a>)
Jv/i
dI(co)
cos d6 ~2tt
f
dl(m)
dd
(14.92)
t/2 dQ. J°° dQ
(remember that 6 is the colatitude). We can estimate the integral for the
lowfrequency range by using the value of the angular distribution (14.87)
at 6 — and the critical angle 6 C (14.89). Then we obtain
dI((o)
I(ca>) ~ 2tt6 c
dQ.
0=0 C\ C
(14.93)
showing that the spectrum increases as co 1  4 for a> < a> c . This gives a very
broad, flat spectrum at frequencies below (o c . For the highfrequency limit
where oj > eo c we can integrate (14.90) over angles to obtain the reasonably
accurate result, 2 , ^
I(co) "JlZryl—) e 2(o/o " (14.94)
c \a> c /
A proper integration of (14.83) over angles yields the expression,
_ 2 f °°
I(co) = 2^3  y — K, A {x) dx (14.95)
c a> P J%(oi<o e
dl(oi)
da
o>«co c
yd ^
Fig. 14.10 Differential frequency
spectrum as a function of angle.
For frequencies comparable to
the critical frequency a> c , the radi
ation is confined to angles of the
order of y 1 . For much smaller
(larger) frequencies, the angular
spread is larger (smaller).
[Sect. 14.6]
Radiation by Moving Charges
487
2.0
1 1 — I l I l I
T 1 1 I II I I
CO/Wc
Fig. 14.11 Synchrotron radiation spectrum (energy radiated per unit frequency
interval) as a function of frequency. The intensity is measured in units of ye 2 /c, while
the frequency is expressed in units of co c (14.85).
In the limit co < co c this reduces to the form (14.93) with a numerical
coefficient 3.25, while for co > eo c it is equal to (14.94). The behavior of
I(co) as a function of frequency is shown in Fig. 14.11. The peak intensity
is of the order of e^yjc, and the total energy is of the order of e 2 ycoJc =
3e 2 y 4 //>. This is in agreement with the value of 47re 2 y 4 /3/o for the radiative
loss per revolution (14.32) in circular accelerators.
The radiation represented by (14.83) and (14.95) is called synchrotron
radiation because it was first observed in electron synchrotrons (1948).
The theoretical results are much older, however, having been obtained for
circular motion by Schott (1912). For periodic circular motion the
spectrum is actually discrete, being composed of frequencies which are
integral multiples of the fundamental frequency co Q = c/p. Since the
charged particle repeats its motion at a rate of c\2ttp revolutions per second,
it is convenient to talk about the angular distribution of power radiated
into the nth multiple of co instead of the energy radiated per unit frequency
interval per passage of the particle. To obtain the harmonic power
expressions we merely multiply 1(a)) (14.95) or dI(oi)jdQ. (14.83) by the
repetition rate cjlirp to convert energy to power, and by co = cjp to
convert per unit frequency interval to per harmonic. Thus
da 277v
P n = — yJI((o = nco )
dljco)
dQ
(14.96)
488 Classical Electrodynamics
These results have been compared with experiment in some detail. * For
this purpose it is necessary to average the spectra over the acceleration
cycle of the machine, since the electron's energy increases continually (see
Problem 14.13). With 80 Mev maximum energy, the spectrum extends
from the fundamental frequency of co ^ 10 9 sec 1 to co c ~ 10 16 sec 1 , or
A ^ 1700 angstroms. The radiation covers the visible region and is bluish
white in color. Careful measurements are in full agreement with theory.
Synchrotron radiation has been observed in the astronomical realm
associated with sunspots, the Crab nebula, and perhaps the ^ dO 3 Mc/sec
radiation from Jupiter. For the Crab nebula the radiation spectrum
extends over a frequency range from radiofrequencies into the extreme
ultraviolet, and shows very strong polarization. From detailed observa
tions it can be concluded that electrons with energies ranging up to 10 12 ev
are emitting synchrotron radiation while moving in circular or helical
orbits in a magnetic induction of the order of 10~ 4 gauss (see Problem
14.15). The radio emission from Jupiter apparently comes from electrons
trapped in Van Allen belts at distances several radii from Jupiter's surface.
Whether these are relativistic electrons emitting synchrotron radiation, or
nonrelativistic electrons emitting socalled cyclotron radiation as they
spiral in the planet's magnetic field, is not clear at present. In any event,
the radiation is strongly polarized parallel to the equator of Jupiter, as
expected for particles trapped in a dipole field and spiraling around lines
of force.
14.7 Thomson Scattering of Radiation
If a plane wave of monochromatic electromagnetic radiation is incident
on a free particle of charge e and mass m, the particle will be accelerated
and so emit radiation. This radiation will be emitted in directions other
than that of the incident plane wave, but for nonrelativistic motion of the
particle it will have the same frequency as the incident radiation. The
whole process may be described as scattering of the incident radiation.
The instantaneous power radiated by a particle of charge e in non
relativistic motion is given by Larmor's formula (14.21),
4E. = J & S in 2 (14.97)
dQ 4ttc 3
where © is the angle between the observation direction and the accelera
tion. The acceleration is provided by the incident plane electromagnetic
* F. R. Elder, R. V. Langmuir, and H. C. Pollock, Phys. Rev., 74, 52 (1948); and
especially D. H. Tomboulain and P. L. Hartman, Phys. Rev., 102, 1423 (1956).
[Sect. 14.7] Radiation by Moving Charges 489
wave. If the propagation vector is k, and the polarization vector c, the
electric field can be written
E(x, = cV^*"" (14.98)
Then, from the force equation for nonrelativistic motion, we have the
acceleration,
v(f) = c  E J k * iwt (14.99)
m
If we assume that the charge moves a negligible part of a wavelength
during one cycle of oscillation, the time average of v 2 is Re (v • v*). Then
the average power per unit solid angle can be expressed as
%)=t E 46Y sin * & (14100)
Since the process is most simply viewed as a scattering, it is convenient to
introduce a scattering cross section, defined by
da Energy radiated/unit time/unit solid angle f14 10n
dQ. ~~ Incident energy flux in energy/unit area/unit time
The incident energy flux is just the timeaveraged Poynting's vector for
the plane wave, namely, c \E \ 2 IStt. Thus from (14.100) we obtain the
differentia] scattering cross section,
^= l^) 2 sin 2 (14.102)
dQ \mcV
If the wave is incident along the z axis with its polarization vector making
an angle of tp with the x axis, as shown in Fig. 14.12, the angular distri
bution is
sin 2 = 1 sin 2 cos 2 (<f>  y) (14.103)
For unpolarized radiation the cross section is given by averaging over the
angle ip. Thus
Kl + cos 2 0) (14.104)
da_ = V \ 2
dQ. \mcV
This is called the Thomson formula for scattering of radiation by a free
charge, and is appropriate for the scattering of Xrays by electrons or
gamma rays by protons. The angular distribution is as shown in Fig. 14.13
by the solid curve. The total scattering cross section, called the Thomson
cross section, is 0/2 \?
°t = t (—. : )' (14.105)
3 \mc /
490
Classical Electrodynamics
The Thomson cross section is equal to 0.665 x 10~ 24 cm 2 for electrons. The
unit of length, e 2 /mc 2 = 2.82 x 10~ 13 cm, is called the classical electron
radius, since a classical distribution of charge totaling the electronic
charge must have a radius of this order if its electrostatic selfenergy is to
equal the electron mass (see Chapter 17).
The classical Thomson result is valid only at low frequencies. For
electrons quantummechanical effects enter importantly when the frequency
a> becomes comparable to mc 2 /h, i.e, when the photon energy hen is
comparable with, or larger than, the particle's rest energy mc 2 . Another
way of looking at this criterion is that we expect quantum effects to appear
if the wavelength of the radiation is of the order of, or smaller than, the
Compton wavelength fr/mc of the particle. At these higher frequencies
the angular distribution becomes peaked in the forward direction as shown
in Fig. 14.13 by the dotted curves, always having, however, the Thomson
value at zero degrees. The total cross section falls below the Thomson
cross section (14.105). The process is then known as Compton scattering,
and for electrons is described theoretically by the KleinNishina formula.
For reference purposes we quote the asymptotic forms of the total cross
section, as given by the KleinNishina formula :
'8tt/ _ 2hco
(e 2 Y
\mcv
+
mc
hco <^ mc 2
mc
hco
. (ihco
 \mc 2
+
(14.106)
hco ;> mc 2
For protons the departures from the Thomson formula occur at photon
Fig. 14.12
[Sect. 14.8]
Radiation by Moving Charges
491
Fig. 14.13 Differential scattering cross section of unpolarized radiation by a free
electron. The solid curve is the classical Thomson result. The dotted curves are given
by the quantummechanical KleinNishina formula. The numbers on the curves refer
to values of hco/mc 2 .
energies above about 100 Mev. This is far below the critical energy
hoi ~ Mc 2 ~ 1 Bev which would be expected in analogy with the electron
Compton effect. The reason is that a proton is not a point particle like
the electron with nothing but electromagnetic interactions, but is a complex
entity having a spreadout charge distribution with a radius of the order of
0.8 x 10~ 13 cm caused by strong interactions with pi mesons. The
departure (a rapid increase in cross section) from Thomson scattering
occurs at photon energies of the order of the rest energy of the pi meson
(140 Mev).
14.8 Scattering of Radiation by Quasifree Charges; Coherent and
Incoherent Scattering
In the scattering of Xrays by atoms the angular distribution (14.104) is
observed at wide angles, at least in light elements. But in the forward
direction the scattering per electron increases rapidly to quite large values
compared to the Thomson cross section. The reason is a coherent addition
of the amplitudes from all electrons. From (14.18) it can be seen that the
492 Classical Electrodynamics
radiation field from a number of free charged particles will be
E « =  Z e >\ D (14.107)
C *—> L ff . Jret
j J
With (14.99) for the acceleration of the typical particle, we find
, = >x(nxc)p
c *—i m.
exp
E„ =
/k • x, — iw I f ^
c / J
R<
(14.108)
In calculating the radiation it is sufficient to approximate 7?, in the
exponent by the form (14.63). Then, in complete analogy with the steps
from (14.97) to (14.102), we find the scattering cross section,
da
d~Q
where
2— 2 A "
q = — n — k
c
sin 2
(14.109)
(14.110)
is the vectorial change in wave number in the scattering.
Equation (14.109) applies to free charged particles instantaneously at
positions x,. Electrons in atoms, for example, are not free. But if the
frequency of the incident radiation is large compared to the characteristic
frequencies of binding, the particles can be treated as free while being
accelerated by a pulse of finite duration. Thus (14.109) can be applied to
the scattering of highfrequency (compared to binding frequencies) radia
tion by bound charged particles. The only thing that remains before
comparison with experiment is to average (14.109) over the positions of all
the particles in the bound system. Thus the observable cross section for
scattering is
da
dQ.
J**,
sin
(14.111)
where the symbol < > means average over all possible values of x 3 .
The cross section (14.111) shows very different behavior, depending on
the value of q. The coordinates x, have magnitudes of the order of the
linear dimensions of the bound system. If we call this dimension a, then
the behavior of the cross section is very different in the two regions,
qa < 1 and qa > 1 . If the scattering angle is 6, the magnitude of q is
2k sin (6/2). Thus the dividing line between the two domains occurs for
[Sect. 14.8] Radiation by Moving Charges 493
angles such that
2kasin~l (14.112)
2
If the frequency is low enough so that ka < 1, then the limit qa < 1 will
apply at all angles. But for frequencies where ka > 1, there will be a region
of forward angles less than
c ~i (14.113)
ka
where the limit qa < 1 holds, and a region of wider angles where the limit
qa > 1 applies.
For qa < 1, the arguments of exponents in (14.1 1 1) are all so small that
the exponential factors can be approximated by unity. Then the differential
cross section becomes
.. da
hm — =
aa>0 dQ.
S ^ 2 sin 2 = Z 2 (—J sin 2 (14.114)
^i m.c 2 \mc 2 /
where the last form is appropriate for electrons in an atom of atomic
number Z. This shows the coherent effect of all the particles, giving an
intensity corresponding to the square of the number of particles times the
intensity for a single particle.
In the opposite limit of qa > 1 the arguments of the exponents are large
and widely different in value. Consequently the cross terms in the square
of the sum will average to zero. Only the absolute squared terms will
survive. Then the cross section takes the form :
Hm S = y (—J sm2 = z (A \f sm2 ° ( i4  ii5 >
i
where again the final form is for electrons in an atom. This result corre
sponds to the incoherent superposition of scattering from the individual
particles.
For the scattering of Xrays by atoms the critical angle (14.113) can be
estimated, using (13.95) as the atomic radius. Then one finds the numerical
value,
6 C ~ — (14.116)
hco (kev)
For angles less than 6 C the cross section rises rapidly to a value of the order
of (14.114), while at wide angles it is given by Z times the Thomson result,
(14.115), or for highfrequency Xrays or gamma rays by the Klein
Nishina formula, shown in Fig. 14.13.
494
Classical Electrodynamics
14.9 Cherenkov Radiation
A charged particle in uniform motion in a straight line in free space does
not radiate. But a particle moving with constant velocity through a material
medium can radiate if its velocity is greater than the phase velocity of light
in the medium. Such radiation is called Cherenkpv radiation, after its
discoverer, P. A. Cherenkov (1937). The emission of Cherenkov radiation
is a cooperative phenomenon involving a large number of atoms of the
medium whose electrons are accelerated by the fields of the passing particle
and so emit radiation. Because of the collective aspects of the process it
is convenient to use the macroscopic concept of a dielectric constant e
rather than the detailed properties of individual atoms.
A qualitative explanation of the effect can be obtained by considering
the fields of the fast particle in the dielectric medium as a function of time.
We denote the velocity of light in the medium by c and the particle velocity
by v . Figure 14.14 shows a succession of spherical field wavelets for v < c
and for v > c. Only for v > c do the wavelets interfere constructively to
form a wake behind the particle. The normal to the wake makes an angle
u< c
Fig. 14.14 Cherenkov radiation. Spherical wavelets of fields of a particle traveling
less than, and greater than, the velocity of light in the medium. For v > c, an electro
magnetic "shock" wave appears, moving in the direction given by the Cherenkov
angle d c .
[Sect. 14.9] Radiation by Moving Charges 495
6 C with the velocity direction, where
cos 6 C =  (14.117)
v
This is the direction of emission of the Cherenkov radiation.
Although we have already found the fields appropriate to the Cherenkov
radiation problem in Section 13.4, and have even given an expression
(13.82) for the energy emitted as Cherenkov radiation, it is instructive to
look at the problem from the point of view of the LienardWiechert
potentials. We will make use of Section 13.4 to the extent of noting that,
for a nonpermeable medium, we may discuss the fields and energy radiated
as if the particle moved in free space with a velocity v > c, provided at the
end of the calculation we make the replacements,
c < , «, L (14.118)
where e is the dielectric constant.* We will simplify the analysis by
assuming that e is independent of frequency. But our final results will be
for individual frequency components and so will be easily generalized.
For a point charge in arbitrary motion the LienardWiechert potentials
were obtained in Section 14. 1 . It was tacitly assumed there that the particle
velocity was less than the velocity of light. Then the potentials (14.6) at a
given point in spacetime depended on the behavior of the particle at one
earlier point in spacetime, the retarded position. This situation corre
sponds in the left side of Fig. 14.14 to the fact that a given point lies on
only one circle. When v > c, however, we see from the right side of the
figure that two retarded positions contribute to the field at a given point in
spacetime. The scalar potential in (14.6) is replaced by
$(x, = e
1
+ e
kRji LkRJ2
1
(14.119)
where the indices 1 and 2 indicate the two retarded times t{ and t 2 '.
To determine the two times t{ and t 2 ' we consider the vanishing of the
argument of the delta function in (14.3):
t > + ' x ~ r ^ t = (14.120)
c
* From (13.54) it is evident that we are dealing in this way with potentials <X>' = Ve®,
A' = A, and fields E' = VcE, B' = B. Then, for example, Poynting's vector is
S' = — (E' X B') ^ — C — (E' XB') = S
4tt 4ttV€
496
Classical Electrodynamics
Fig. 14.15
For a particle with constant velocity v, we can take r(t') = yf. With
X = (x — vt) as the vector distance from the present position P' of the
particle to the observation point P, (14.120) becomes
(t  t') =  X + y(t  t')\
The solution of this quadratic is
(t  f) =
X • v ± V(X . v) 2  (v 2  c 2 )X 2
(u 2  c 2 )
(14.121)
(14.122)
Only roots that are real and positive have physical meaning. For v < c,
the square root is real and larger than X • v. Hence there is only one
valid root for (t — t'), as already noted. But for v > c, there are other
possibilities. First we note that even when the square root is real (as it is for
directions more or less parallel or antiparallel to v) it is smaller in magni
tude than X • v. Consequently there is no root for (t — t') when X and v
have an acute angle between them; the fields do not get ahead of the
particle. If a is the angle between X and v, as shown in Fig. 14.15, we see
furthermore that the square root is imaginary for cos 2 a < [1 — (c 2 /v 2 )].
But for backward angles, such that
cos
< a < 77
there are two real, positive values of (t — t') as solutions of (14.121). Thus
the potentials exist only inside the Cherenkov cone defined by cos a =
[1 (c 2 !v 2 )p.
[Sect. 14.9] Radiation by Moving Charges 491
The values of kR corresponding to the two roots (14.122) are easily
shown to be
K R = T  [(X • v) 2  (v*  c 2 )X 2 ] H (14.123)
c
Actually in (14.119) the absolute values are required, because of the sign
inherent in the Jacobian derivative in (14.4). Thus the two terms add, and
the potentials can be written
<D(x, =
Xy/l  (v 2 lc 2 ) sin 2 a
A(x, = ^>(x,
(14.124)
c
These potentials are valid inside the Cherenkov cone, become singular on
its surface, and vanish outside the cone. They represent a wave front
traveling at velocity c in the direction d c (14.1 17). The singularity is not a
physical reality, of course. It comes from our assumption that the velocity
of light in the medium is independent of frequency. For high enough
frequencies (short enough wavelengths) the phase velocity of light in the
medium will approach the velocity of light in vacuo. This variation with
frequency will cause a smoothing at short distances which will eliminate
singularities.
The potentials (14.124) are special cases of the potentials whose Fourier
transforms are given by (13.57). The fields which can be found from
(14.124) are similarly the Fourier transforms of the fields (13.64) and
(13.65), assuming e(eo) is a real constant. The calculation of energy
radiated proceeds exactly as in Section 13.4 with the integration of the
Poynting's vector over a cylinder, as in (13.71), yielding the final expression
(13.82) for energy radiated per unit distance.
The discussion presented so far, with the appearance of a Cherenkov
"shock wave" for v > c, is the proper macroscopic description of the
origin of Cherenkov radiation. If, however, one is interested only in the
angular and frequency distribution of the radiation and not in the mecha
nism, it is possible to give a simple, nonrigorous derivation, using the
substitutions ( 1 4. 1 1 8). The angular and frequency distribution of radiation
emitted by a charged particle in motion is given by (14.67):
dI(co) e 2 co 2 If 00 , , to (' 5 T Q )
— ^^ = n x (n x \)e \ c '
For a particle moving in a nonpermeable, dielectric medium transfor
mation (14.118) yields
— ^—^ = e^ n x (n x \)e \ ° ' dt (14.126)
dQ. 47r 2 c 3 I J°°
(14.125)
498 Classical Electrodynamics
For a uniform motion in a straight line, r(t) = vt. Then we obtain
dl(w)
n x v
o>_ f°° i<oth
1lT J oo
dQ. c
The integral is a Dirac delta function. Then
dI(co) e 2 e A p 2 sin 2
v ) dt
dQ.
1 6(1  e /2 /? cos 0) s
(14.127)
(14.128)
where 6 is measured relative to the velocity v. The presence of the delta
function guarantees that the radiation is emitted only at the Cherenkov
angle 6 C :
cos 6 C = y (14.129)
The presence of the square of a delta function in angles in (14.128) means
that the total energy radiated per unit frequency interval is infinite. This
infinity occurs because the particle has been moving through the medium
forever. To obtain a meaningful result we assume that the particle passes
through a slab of dielectric in a time interval IT. Then the infinite integral
in (14.127) is replaced by
jo f y e «(i  g") dt = ^ Sin ^ r(1  ^ C ° S °>3 (14.130)
2ttJt tt [coT(1  e A p cos 0)]
The absolute square of this function peaks sharply at the angle 6 C , provided
coT > 1. Assuming that /? > —^ , so that the angle d c exists, the integral
over angles is
'^/ojiy sin>[a>T(l^cosOX = 2joT
\ rr 1 [coT(l  e*P COS 0)] 2 fie*
showing that the amount of radiation is proportional to the time interval.
From (14.128) we find that the total energy radiated per unit frequency
interval in passing through the slab is
/■
1(c) = tE S i n 2 Q c ( 2cj 8T)
c 2
(14.132)
This can be converted into energy radiated per unit frequency interval per
unit path length by dividing by 2cfiT. Then, with (14. 129) for 6 C , we obtain
dI((o)
dx
e co
^2
1
1
P 2 e(co)J
(14.133)
where co is such that e(co) > (l//? 2 ), in agreement with (13.82).
[Sect. 14.9]
Radiation by Moving Charges
499
€(W)
Fig. 14.16 Cherenkov band.
Radiation is emitted only in
shaded frequency range where
e(co) > jS 2 .
The properties of Cherenkov radiation can be utilized to measure
velocities of fast particles. If the particles of a given velocity pass through
a medium of known dielectric constant e, the light is emitted at the
Cherenkov angle (14.129). Thus a measurement of the angle allows deter
mination of the velocity. Since the dielectric constant of a medium in
general varies with frequency, light of different colors is emitted at some
what different angles. Figure 14.16 shows a typical curve of e(eo), with a
region of anomalous dispersion at the upper end of the frequency interval.
The shaded region indicates the frequency range of the Cherenkov light.
Since the dielectric medium is strongly absorbent at the region of anom
alous dispersion, the escaping light will be centered somewhat below the
resonance. Narrow band filters may be employed to select a small interval
of frequency and so improve the precision of velocity measurement. For
very fast particles (£ < 1) a gas may be used to provide a dielectric
constant differing only slightly from unity and having (e — 1) variable
over wide limits by varying the gas pressure. Counting devices using
Cherenkov radiation are employed extensively in highenergy physics, as
instruments for velocity measurements, as mass analyzers when combined
with momentum analysis, and as discriminators against unwanted slow
particles.
REFERENCES AND SUGGESTED READING
The radiation by accelerated charges is at least touched on in all electrodynamics
textbooks, although the emphasis varies considerably. The relativistic aspects are treated
in more or less detail in
Iwanenko and Sokolow, Sections 3943,
Landau and Lifshitz, Classical Theory of Fields, Chapters 8 and 9,
Panofsky and Phillips, Chapters 18 and 19,
Sommerfeld, Electrodynamics, Sections 29 and 30.
Extensive calculations of the radiation emitted by relativistic particles, anticipating many
results rederived in the period 19401950, are presented in the interesting monograph by
Schott.
500 Classical Electrodynamics
The scattering of radiation by charged particles is presented clearly by
Landau and Lifshitz, Classical Theory of Fields, Sections 9.119.13, and
Electrodynamics of Continuous Media, Chapters XIV and XV.
PROBLEMS
14.1 Verify by explicit calculation that the LienardWiechert expressions for all
components of E and B for a particle moving with constant velocity agree
with the ones obtained in the text by means of a Lorentz transformation.
Follow the general method at the end of Section 14.1.
14.2 Using the LienardWiechert fields, discuss the timeaverage power radiated
per unit solid angle in nonrelativistic motion of a particle with charge e,
moving
(a) along the z axis with instantaneous position z(t) = a cos co t,
(b) in a circle of radius R in the xy plane with constant angular
frequency co .
Sketch the angular distribution of the radiation and determine the total
power radiated in each case.
14.3 A nonrelativistic particle of charge ze, mass m, and kinetic energy E makes
a headon collision with a fixed central force field of finite range. The
interaction is repulsive and described by a potential V(r), which becomes
greater than E at close distances.
(a) Show that the total energy radiated is given by
N 2 Jr mill
3 ra 2 c 3 .
dV
dr
Vv(r m m)  V{r)
where r m in is the closest distance of approach in the collision.
(b) If the interaction is a Coulomb potential V(r) = zZe 2 jr, show that
the total energy radiated is
AlV =
zmv 6
45 Zc 3
where v is the velocity of the charge at infinity.
14.4 A particle of mass m, charge q, moves in a plane perpendicular to a
uniform, static, magnetic induction B.
(a) Calculate the total energy radiated per unit time, expressing it in
terms of the constants already defined and the ratio y of the particle's
total energy to its rest energy.
(b) If at time t = the particle has a total energy E = y Q m&, show
that it will have energy E = ymc 2 < E at a time t, where
f _ 3m 3 c 5 /I _ 1
_ 2(fB 2 \y Yo)
provided y > 1 .
(c) If the particle is initially nonrelativistic and has a kinetic energy €
at t = 0, what is its kinetic energy at time tl
id) If the particle is actually trapped in the magnetic dipole field of the
earth and is spiraling back and forth along a line of force, does it radiate
[Probs. 14]
Radiation by Moving Charges
501
14.5
14.6
14.7
14.8
more energy while near the equator, or while near its turning points?
Why? Make quantitative statements if you can.
As in Problem 14.2a a charge e moves in simple harmonic motion along
the z axis, z(t') = a cos (oo t').
{a) Show that the instantaneous power radiated per unit solid angle is:
dP(t') _ e 2 cft 4 sin 2 6 cos 2 (cot')
dQ. 477a 2 (1 + p cos sin co t'f
where P = acojc.
(b) By performing a time averaging, show that the average power per
unit solid angle is :
JD „2„/?4 r A i R2 „™2 A 1
sin 2
(c) Make rough sketches of the angular distribution for nonrelativistic
and relativistic motion.
Show explicitly by use of the Poisson sum formula or other means that,
if the motion of a radiating particle repeats itself with periodicity T, the
continuous frequency spectrum becomes a discrete spectrum containing
frequencies that are integral multiples of the fundamental. Show that a
general expression for the power radiated per unit solid angle in each
multiple m of the fundamental frequency co = 2tt \T is:
dP
e 2 cp
4 + p 2 cos 2 e
dCl
327ra 2
_(i  p 2 cos 2 eyA_
dP m
dCl
? 2 (o^m 2
1 '
v(/) x nexp
4^)]"'
(a) Show that for the simple harmonic motion of a charge discussed in
Problem 14.5 the average power radiated per unit solid angle in the mth
harmonic is :
dP e 2 cB 2
ii§ = S m% tan2 QJmKm ^ cos 6)
(b) Show that in the nonrelativistic limit the total power radiated is all
in the fundamental and has the value:
2e 2
33^
4„2
where a 2 is the mean square amplitude of oscillation.
A particle of charge e moves in a circular path of radius R in the xy plane
with constant angular velocity o> .
(a) Show that the exact expression for the angular distribution of power
radiated into the mth multiple of oo is :
dPrn
dCl
e 2 <o 4 i? 2
27TC 3
,n dJ m (mP sin d) \ 2
\ d(mP sin 6) J
cot 2 6
+ 35 Jm\mP sin0)
where p = co Q Rlc, and J m (x) is the Bessel function of order m.
(b) Assume nonrelativistic motion and obtain an approximate result for
dP m /dCl. Show that the results of Problem 14.26 are obtained in this limit.
(c) Assume extreme relativistic motion and obtain the results found in
the text for a relativistic particle in instantaneously circular motion.
(Watson, pp. 79, 249, may be of assistance to you.)
502 Classical Electrodynamics
14.9 Bohr's correspondence principle states that in the limit of large quantum
numbers the classical power radiated in the fundamental is equal to the
product of the quantum energy (Ha> ) and the reciprocal mean lifetime of
the transition from principal quantum number n to (n — 1).
(a) Using nonrelativistic approximations, show that in a hydrogenlike
atom the transition probability (reciprocal mean lifetime) for a transition
from a circular orbit of principal quantum number n to (n — 1) is given
classically by
1 = 2£7Z£YwcM
t 3 fic\ he J h n b
(b) For hydrogen compare the classical value from (a) with the correct
quantummechanical results for the transitions 2p + Is (1.6 x 10 _9 sec),
4/^3^/(7.3 x 10 8 sec), 6/* —5^ (6.1 x 10 7 sec).
14.10 Periodic motion of charges gives rise to a discrete frequency spectrum in
multiples of the basic frequency of the motion. Appreciable radiation in
multiples of the fundamental can occur because of relativistic effects
(Problems 14.7 and 14.8) even though the components of velocity are truly
sinusoidal, or it can occur if the components of velocity are not sinusoidal,
even though periodic. An example of this latter motion is an electron
undergoing nonrelativistic elliptic motion in a hydrogen atom.
The orbit can be specified by the parametric equations
x — a(cos u — e)
y = aV\ — e 2 sin u
where
co t = u — c sin u
a is the semimajor axis, e is the eccentricity, co is the orbital frequency,
and u is the angle in the plane of the orbit. In terms of the binding energy
B and angular momentum L, the various constants are
e 2 L 2BL* 8B S
IB' V me 4 ' ° me*
(a) Show that the power radiated in the &th multiple of co is
*£ (to Hp
(J k '(ke)f + \l—f\j k \ke) j
where J k (x) is a Bessel function of order k.
(b) Verify that for circular orbits the general result (a) agrees with part
(a) of Problem 14.9.
14.11 Instead of a single charge e moving with constant velocity co in a circular
path of radius R, as in Problem 14.8, a set of N such charge moves with
fixed relative positions around the same circle.
(a) Show that the power radiated into the wth multiple of co is
dPJJN) dPJX) F fm
d£l dCl m{ }
[Probs. 14] Radiation by Moving Charges 503
where dP m {\)ldO. is the result of part (a) in Problem 14.8, and
F m (N)
N
2^
i=i
Bj being the angular position of they'th charge at t = t .
(b) Show that, if the charges are uniformly spaced around the circle,
energy is radiated only into multiples of Na> , but with an intensity N 2
times that for a single charge. Give a qualitative explanation of these facts.
(c) Without detailed calculations show that for nonrelativistic motion
the dependence on N of the total power radiated is dominantly as fi 2N , so
that in the limit N »■ oo no radiation is emitted.
(d) By arguments like those of (c) show that for relativistic particles the
radiated power varies with N mainly as exp (— 27V/3y 3 ) for N > y 3 , so
that again in the limit N + <x> no radiation is emitted.
(e) What relevance have the results of (c) and (d) to the radiation
properties of a steady current in a loop ?
14.12 As an idealization of steadystate currents flowing in a circuit, consider a
system of N identical charges q moving with constant speed v (but subject
to accelerations) in an arbitrary closed path. Successive charges are
separated by a constant small interval A.
Starting with the LienardWiechert fields for each particle, and making
no assumptions concerning the speed v relative to the velocity of light,
show that, in the limit N * oo, q >0, and A >0, but Nq = constant
and q/A = constant, no radiation is emitted by the system and the electric
and magnetic fields of the system are the usual static values.
(Note that for a real circuit the stationary positive ions in the conductors
will produce an electric field which just cancels that due to the moving
charges.)
14.13 Assume that the instantaneous power spectrum radiated by an electron in
a synchrotron is given by
P(o>, t) ~   y{t) (—) %*»!<»,
77 P \€0j
where co c = 3to y 3 (/).
(a) If the electrons increase their energy approximately linearly during
one cycle of operation, show that the power spectrum, averaged over one
cycle of operation, is
(P(co, /)> ^t Vm& yA r^dy
3* p Jx y A
where x = 2o>/a> cmax
(b) Determine limiting forms for the spectrum when x < 1 and x > 1 .
(c) By finding tables of the integral (it is an incomplete gamma function)
or by graphical integration for x = 0. 1 , 0.5, 1 .0, 1.5, determine numerically
the spectrum, plot it as a function of log [ft>/eo cmax ], and compare it with
the curves given by Elder, Langmuir, and Pollock, Phys. Rev., 74, 52
(1948), Fig. 1.
14.14 (a) Within the framework of approximations of Section 14.6, show that,
for a relativistic particle moving in a path with instantaneous radius of
curvature />, the frequencyangle spectra of radiations with positive and
504 Classical Electrodynamics
negative helicity are
dl
wum+'l
*w«)±7; —Knit)
(b) From the formulas of Section 14.6 and (a) above, discuss the
polarization of the total radiation emitted as a function of frequency and
angle. In particular, determine the state of polarization at (1) high
frequencies (co > co c ) for all angles, (2) intermediate and low frequencies
(co < co c ) for large angles, (3) intermediate and low frequencies at very
small angles.
(c) See the paper by P. Joos, Phys. Rev. Letters, 4, 558 (1960), for
experimental comparison.
14.15 Consider the synchrotron radiation from the Crab nebula. Electrons with
energies up to at least 10 12 ev move in a magnetic field of the order of
10~ 4 gauss.
(a) For E = 10 12 ev, B = 10 4 gauss, calculate the orbit radius p, the
fundamental frequency a> = c/p, and the critical frequency co c .
(b) Show that for a relativistic electron of energy £ in a constant
magnetic field the power spectrum of synchrotron radiation can be written
tf'te)
P(E, co) = const
where f(x) is a cutoff function having the value unity at * = and
vanishing rapidly for x > 1 [e.g.,/ ~ exp ( 2co/co c ), as in Problem 14.13],
and to c = (3eB/mc)(Elmc 2 ) 2 cos 6, where 6 is the pitch angle of the helical
path.
(c) If electrons are distributed in energy according to the spectrum
N(E) dE ~ E~ n dE, show that the synchrotron radiation has the power
spectrum
<P(co)> dco ~ co~ a dco
where a = (« — l)/2.
(d) Observations on the radiofrequency and optical continuous
spectrum from the Crab nebula show that on the frequency interval from
co ~ 10 8 sec 1 to co ~ 6 x 10 15 sec 1 the constant a ~ 0.35. At higher
frequencies the spectrum of radiation falls steeply with a ^ 1.5. Determine
the index n for the electronenergy spectrum, as well as an upper cutoff for
that spectrum. Is this cutoff consistent with the numbers of part (a) ?
0) From the result of Problem 14.46 find a numerical value for the
time taken by an electron to decrease in energy from infinite energy to
10 12 ev in a field of 10 4 gauss. How does this compare with the known
lifetime of the Crab nebula ?
14.16 Assuming that Plexiglas or Lucite has an index of refraction of 1.50 in the
visible region, compute the angle of emission of visible Cherenkov
radiation for electrons and protons as a function of their energies in Mev.
Determine how many quanta with wavelengths between 4000 and 6000
angstroms are emitted per centimeter of path in Lucite by a 1Mev
electron, a 500Mev proton, a 5Bev proton.
15
Bremsstrahlung ,
Method of Virtual Quanta,
Radiative Beta Processes
In Chapter 14 radiation by accelerated charges was discussed in a
general way, formulas were derived for frequency and angular distributions,
and examples of radiation by both nonrelativistic and relativistic charged
particles in external fields were treated. The present chapter is devoted to
problems of emission of electromagnetic radiation by charged particles in
atomic and nuclear processes.
Particles passing through matter are scattered and lose energy by
collisions, as described in detail in Chapter 13. In these collisions the
particles undergo acceleration ; hence they emit electromagnetic radiation.
The radiation emitted during atomic collisions is customarily called
bremsstrahlung (braking radiation) because it was first observed when high
energy electrons were stopped in a thick metallic target. For nonrelativistic
particles the loss of energy by radiation is negligible compared with the
collisional energy loss, but for ultrarelativistic particles radiation can be
the dominant mode of energy loss. Our discussion of bremsstrahlung and
related topics will begin with the nonrelativistic, classical situation. Then
semiclassical arguments will be used, as in Chapter 13, to obtain plausible
quantummechanical modifications. Relativistic effects, which produce
significant changes in the results, will then be presented.
The creation or annihilation of charged particles is another process in
which radiation is emitted. Such processes are purely quantum mechanical
in origin. There can be no attempt at a classical explanation of the
basic phenomena. But given that the process does occur, we may legiti
mately ask about the spectrum and intensity of electromagnetic radiation
505
506 Classical Electrodynamics
accompanying it. The sudden creation of a fast electron in nuclear beta
decay, for example, can be viewed for our purposes as the violent accelera
tion of a charged particle initially at rest to some final velocity in a very
short time interval, or, alternatively, as the sudden switching on of the
charge of the moving particle in the same short interval. We will discuss
nuclear beta decay and orbitalelectron capture in these terms in Sections
15.7 and 15.8.
In radiation problems, such as the emission of bremsstrahlung or radia
tive beta decay, the wave nature of the charged particles involved produces
quantummechanical modifications very similar to those appearing in our
earlier energyloss considerations. These can be taken into account in a
relatively simple way. But there is a more serious deficiency which occurs
only in radiation problems. It is very difficult to take into account the
effects on the trajectory of the particle of the energy and momentum carried
off by radiation. This is not only because radiation reaction effects are
relatively hard to include (see Chapter 17), but also because of the discrete
quantum nature of the photons emitted. Thus, even when modifications
are made to describe the quantummechanical nature of the particles, our
results are limited in validity by the restriction that the emitted photon have
an energy small compared to the total energy available. At the upper end
of the frequency spectrum our semiclassical expressions will generally have
only qualitative validity.
15.1 Radiation Emitted during Collisions
If a charged particle makes a collision, it undergoes acceleration and
emits radiation. If its collision partner is also a charged particle, they both
emit radiation and a coherent superposition of the radiation fields must be
made. Since the amplitude of the radiation fields depends (nonrelativisti
cally) on the charge times the acceleration, the lighter particle will radiate
more, provided the charges are not too dissimilar. In many applications
the mass of one collision partner is much greater than the mass of the other.
Then for the emission of radiation it is sufficient to treat the collision as the
interaction of the lighter of the two particles with a fixed field of force. We
will consider only this situation, and will leave more involved cases to the
problems at the end of the chapter.
From (14.65) we see that a nonrelatlvistic particle with charge e and
acceleration c(3(f) radiates energy with an intensity per unit solid angle per
unit frequency interval,
dI(co) e 2 [ , . Ut™™) 2
[Sect. 15.1] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 507
The position vector r(/) has the order of magnitude (v)t, relative to a suit
able origin, where (v) is a typical velocity of the problem. This means that
the second term in the exponential in (15.1) is of the order of (v)/c times
the first. For nonrelativistic motion, it can be neglected. Its neglect is
sometimes called the dipole approximation, in analogy with the multipole
expansion of Section 9.2. Then we find the approximate expression,
dljco) = e 2
dO. 4tt 2 c
I
n x (n x $)e imt dt
(15.2)
If we consider a collision process, the acceleration caused by the field of
force exists only for a limited time t, namely, the collision time :
r~ (15.3)
v
where a is a characteristic distance over which the force is appreciable.
Then the integral in (15.2) is over a time interval of order t. This means
that t provides a natural parameter with which to divide the frequencies of
the radiation emitted into low frequencies (cot < 1) and high frequencies
(cot > 1). In the lowfrequency limit, the exponential in (15.2) is sensibly
constant over the period of acceleration. Then the integration can be
performed immediately :
(p(t)e i(at dt ~ p(0 dt = p 2  Pi = Ap (15.4)
where c$ x and c(J 2 are the initial and final velocities, and A(S is the
vectorial change. Then the energy radiated is
^ } ~ —  API 2 sin 2 0, cot < 1 (15.5)
dn 4tt 2 c ^
where is measured relative to the direction of A (5. The total energy
radiated per unit frequency interval in this limit is
I(co) ~ —  API 2 , cot < 1 (15.6)
3tt c
In the highfrequency limit (cot > 1) the exponential in (15.2) oscillates
very rapidly compared to the variation of $(t) in time. Consequently the
integrand has a very small average value, and the energy radiated is
negligible. The frequency spectrum will appear qualitatively as shown in
Fig. 15.1. It will be convenient sometimes to make the approximation that
508 Classical Electrodynamics
the spectrum is given by a step function :
' 2e 2
/(*>) =
I API a»<l
0, COT > 1
For a single encounter with a definite A3 this is not a very good approxi
mation, but if an average over many collisions with various A (3 is wanted
the approximation is adequate.
The angular distribution (15.5) includes all polarizations of the emitted
radiation. Sometimes it is of interest to exhibit the intensity for a definite
state of polarization. In collision problems it is usual that the direction of
the incident particle is known and the direction of the radiation is known,
but the deflected particle's direction, and consequently that of A (3, are not
known. Consequently the plane containing the incident beam direction
and the direction of the radiation is a natural one with respect to which
one specifies the state of polarization of the radiation.
For simplicity we consider a small angle deflection so that A (3 is approxi
mately perpendicular to the incident direction. Figure 15.2 shows the
vectorial relationships. Without loss of generality n, the observation
direction, is chosen in the xz plane, making an angle 6 with the incident
beam. The change in velocity A (3 lies in the xy plane, making an angle cf>
with the x axis. Since the direction of the scattered particle is not observed,
we will average over j>. The unit vectors €,, and e ± are polarization vectors
parallel and perpendicular to the plane containing (3 X and n.
The direction of polarization of the radiation is given by the vector
n x (n x A (3). This is perpendicular to n (as it must be) and can be
resolved into components along e,, and e ± . Thus
n x (n x A(3) = A0[cos d cos <j> e„ — sin <f> € ± ] (15.8)
The absolute squares of the components in (15.8), averaged over <f>, give
I(O)^\A0\
1/7
Fig. 15.1 Frequency spectrum of radiation emitted in
a collision of duration t with velocity change A/J.
[Sect. 15.2] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 509
Fig. 15.2
the intensities of radiation for the two polarization states. The results are
dD.
dI L ((o)
dQ.
8tt 2 c
_£!_
8tt 2 c
= VI Ap 2 cos 2
A(J<
(15.9)
These angular distributions are valid for all types of nonrelativistic small
angle collisions. They have been verified in detail for the continuous
Xray spectrum produced by electrons of kinetic energies in the kilovolt
range. It is evident that the sum of the two intensities is consistent with
(15.5) and yields a total radiated intensity equal to (15.6).
15.2 Bremsstrahlung in Nonrelativistic Coulomb Collisions
The most common situation where a continuum of radiation is emitted
is the collision of a fast particle with an atom. As a model for this process
we will consider first the collision of a fast, but nonrelativistic, particle of
charge ze, mass M, and velocity v with a fixed point charge Ze. For
simplicity we will assume that the deflection of the incident particle is small.
Then the same arguments as were used in Chapter 13 on the limits of impact
parameters will be involved. In fact, much of the discussion can be trans
planted bodily from the treatment of energy loss.
For a small deflection in a point Coulomb field of charge Ze the
momentum change is transverse and is given by (13.1) times Z. Thus the
net change of velocity of the incident particle passing at impact parameter
b has the magnitude
Ay =
2zZe 2
Mvb
(15.10)
510 Classical Electrodynamics
The frequency spectrum will be given approximately by (15.7) (times z 2 )
with a collision time (15.3) r~ b/v. Thus the spectrum extends from
co = to comax ^ v/b :
' 8 (z 2 e 2 \ 2 Z 2 e 2
37rWc 2 / c
/(co, b) ~
0,
1
ft 2 '
V
CO < —
ft
co > 
ft
(15.11)
Just as in the energyloss process, the useful physical quantity is a cross
section obtained by integrating over all possible impact parameters.
Accordingly we define the radiation cross section %{co), with dimensions
(areaenergy/frequency),
x(co) = J /(co, ft)277ft db (15.12)
The classical limits on the impact parameters can be found by arguments
analogous to those of Section 13.1. The classical minimum impact param
eter is [see(13.5)(13.7)]:
^ (c) _ zZe 2
Mv 2
ftmin = . (15.13)
while the maximum value is governed by the cutoff in the spectrum (15. 1 1).
If we fix our attention on a given frequency co in %(co), it is evident that only
for impact parameters less than
ftmax^ (15.14)
ft)
will the accelerations be violent and rapid enough to produce significant
radiation at that frequency. With these limits on ft the radiation cross
section is
where A is a numerical factor of the order of unity which takes into account
our uncertainties in the exact limits on impact parameters. This result is
valid only for frequencies where the argument of the logarithm is large
compared to unity, corresponding to ft max > ft min . This means that there
is a classical upper limit co ( ^ x to the frequency spectrum given by
co£L (15.16)
zZe*
For highly charged, massive, slow particles the classical radiation cross
section will be valid, but just as in the energyloss phenomena the wave
[Sect. 15.2] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 511
nature of particles enters importantly for lightly charged, swift particles.
The quantum modifications are very similar to those discussed in Section
13.3. The wave nature of the incident particle sets a quantummechanical
lower limit on the impact parameters,
b8L~— (15.17)
Mv
This means that the radiation cross section is approximately
instead of (15.15). We note that the arguments of the logarithm differ by
the factor r\ (13.42) (times Z to give the product of the charges). The same
rules about domains of validity of the classical and quantummechanical
formulas apply here as for the energy loss. The frequency spectrum of the
quantum cross section extends up to a maximum frequency co^ x of the
order of
a>SL~^ (15.19)
n
We note that this is approximately the conservation of energy limit,
Wmax = Mv 2 /2h. Since the classical result holds only when r\ > 1, we see
that
a&L ^ " a&L < *>mL (1520)
This shows that the classical frequency spectrum is always confined to very
low frequencies compared to the maximum allowed by conservation of
energy. Thus the classical domain is of little interest. In what follows we
will concentrate on the quantummechanical results.
Although the upper limit (15.19) is in rough accord with conservation
of energy, the quantum radiation cross section has only qualitative validity
at the upper end of the frequency spectrum. As was discussed in the
introduction to this chapter, the reason is the discrete quantum nature of
the photons emitted. For soft photons with energies far from the maximum
the discrete nature is unimportant because the energy and momentum
carried off are negligible. But for hard photons near the end point of the
spectrum the effects are considerable. One obvious and plausible way to
include the conservation of energy requirement is to argue that the impact
parameters (15.14) and (15.17) should involve an average velocity,
(v) = K»* + »/) = "7= (y/E + V^M (1521)
512
Classical Electrodynamics
1
1 1 1
l'i
1

A

3
Sv » v> ^ Semiclassical

X
V Classical
s. ^Z" quantum
Bethels.
>
V^tj = 10
Heitler ^v
—
1
1 Nsj
1

0.2
0.4
0.6
0.8
1.0
■ flu
E
Fig. 15.3 Radiation cross section (energy x area/unit frequency) for Coulomb
collisions as a function of frequency in units of the maximum frequency (E/h). The
classical spectrum is confined to very low frequencies. The curve marked "Bethe
Heitler" is the quantummechanical Born approximation result, while the "semi
classical quantum" curve is (15.18).
where E = \Mv 2 is the initial kinetic energy of the particle, and ho is the
energy of the photon emitted. With this average velocity in place of v in
(15.18), we obtain
X<M —
16 ZV
f z 2 e 2 \ 2 (c\ 2
\McV \v)
I (JE + JE  hoSf
2 ha)
(15.22)
If X = 2, this cross section is exactly the quantummechanical result in
the Born approximation, first calculated by Bethe and Heitler (1934). The
argument of the logarithm evidently equals unity when hoy = \Mv l , so
that the conservation of energy requirement is properly satisfied. Figure
15.3 shows the shape of the radiation cross section as a function of
frequency. The BetheHeitler formula (15.22) is compared with our
classical and semiclassical quantum formulas (15.15) and (15.18) with A = 2
and r\ = 10.
The bremsstrahlung spectrum is sometimes expressed as a cross section
for photon emission with dimensions of area/unit photon energy. Thus
hojo hiems (hco) d(hco) = %{(jo) dco
(15.23)
[Sect. 15.3] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 513
The bremsstrahlung photon cross section is evidently
, t , 16ZV/zV\ 2 /c\ 2 ln( ) n ._
. brems (M^ T — (—)() ^^ (15.24)
where the argument of the logarithm is that of (15.15) or (15.22). Since the
logarithm varies relatively slowly with photon energy, the main dependence
of the cross section on photon energy is as (/ko) _1 , known as the typical
bremsstrahlung spectrum.
The radiation cross section %((o) depends on the properties of the particles
involved in the collision as Z¥/A/ 2 , showing that the emission of radiation
is most important for electrons in materials of high atomic number. The
total energy lost in radiation by a particle traversing unit thickness of
matter containing N fixed charges Ze (atomic nuclei) per unit volume is
Af . /''"max
™* = N %{a>)da> (15.25)
dx Jo
Using (15.22) for %{(o) and converting to the variable of integration
x = (ha)/E), we can write the radiative energy loss as
*&« = 16 NZ (Zl)^ p in (l±JT=*\ dx (15>26)
dx 3 \Hc/Mc 2 Jo \ y/z /
The dimensionless integral has the value unity. For comparison we write
the ratio of radiative energy loss to collision energy loss (13.13) or (13.44):
rf£rad_. 4^mjVj 2 J_ (152?)
i — i
dEcoii 3tt 137 M\c! In J3,
For nonrelativistic particles (v < c) the radiative loss is completely negli
gible compared to the collision loss. The fine structure constant (e 2 /hc =
1/137) enters characteristically whenever there is emission of radiation as
an additional step beyond the basic process (here the deflection of the
particle in the nuclear Coulomb field). The factor m\M appears because
the radiative loss involves the acceleration of the incident particle, while
the collision loss involves the acceleration of an electron.
15.3 Relativistic Bremsstrahlung
For relativistic particles making collisions with atomic nuclei there are
characteristic modifications in the radiation emitted. Our first thought
would be that the nonrelativistic discussion of the previous sections would
not be valid at all, and that a full relativistic treatment would be necessary.
514
Classical Electrodynamics
Laboratory
system
Coordinate
system K'
Fig. 15.4 Radiation emitted during relativistic collisions viewed from the laboratory
(nucleus at rest) and the frame K' (incident particle essentially at rest).
But it is one of the great virtues of the special theory of relativity (aside
from being correct and necessary) that it allows us to choose a convenient
reference frame for our calculation and then transform to the laboratory at
the end. Thus we will find that all but the final steps of the relativistic
bremsstrahlung calculation can be done nonrelativistically.
There are two aspects. First of all, we know that radiation emitted by a
highly relativistic particle is confined to a narrow cone of halfangle of the
order of Mc 2 /E, where E is its total energy. Thus, unless we are interested
in very fine details, it is sufficient to consider only the total energy radiated
at a given frequency. The second point is that except for very close
collisions the incident particle is deflected only slightly in an encounter and
loses only a very small amount of energy. In the reference frame K', where
the incident particle is at rest initially and the nucleus moves by with
velocity v ~ c, the corresponding motion of the incident particle is non
relativistic throughout the collision. This means that in the frame K' the
radiation process can be treated entirely nonrelativistically. The connection
between the radiation process as viewed in the laboratory and in the
coordinate frame K' is sketched in Fig. 15.4.
Almost all the arguments previously presented in Sections 15.1 and 15.2
apply. The only modifications are in the limits on the impact parameters.
The relativistic contraction of the fields (see Section 11.10) makes the col
lision time (11.120) smaller by a factor y = E/Mc 2 . This means that the
maximum impact parameter is increased from (15.14) to
h ~Z?
f max — —
(15.28)
where a>' is the emitted frequency in the coordinate system K'. The
minimum impact parameter for these radiation problems is not the
expected hjp = fr/yMv, even though this is the magnitude of "smearing
[Sect. 15.3] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 515
out" of the particle due to quantum effects. The proper value is still (15.17),
without factors of y, as can be seen from the following argument. In the
emission of radiation all parts of an extended charge distribution must
experience the same acceleration at the same time. Otherwise interference
effects will greatly reduce the amount of radiation. This means that
appreciable radiation will occur only if the width of the pulse of accelera
tion due to the passing nuclear field is large compared to the "smearing
out" of the charge. The pulse width is of the order of b/y, while the
transverse smearingout distance is of the order of hjyMv. This sets a
lower limit on impact parameters equal to (15.17), even for relativistic
motion. With (15.11), (15.12), and these revised impact parameters, the
radiation cross section %'{co') in the system K' is
16 ZV fW \ 2 (cY ln UyMv 2 \
\McV V n \ hco' I
«^~(tt;I () ln r^J ( 15  29 >
To transform this result to the (unprimed) laboratory frame we need to
know the transformation properties of the radiation cross section and the
frequency. The radiation cross section has the dimensions of (cross
sectional area) • (energy) • (frequency) 1 . Since energy and frequency
transform in the same way under Lorentz transformations, while transverse
dimensions are invariant, the radiation cross section is a Lorentz invariant:
X (a>) = X '(co') (15.30)
The transformation of frequencies is according to the relativistic Doppler
shift formula (11.38):
co = yco'(l + P cos 0') (15.31)
where 6' is the angle of emission in the frame K'. The cross section %'(co')
is the total cross section, integrated over angles in K '. Since the accelera
tion is predominantly transverse in that frame, the radiation is emitted
essentially symmetrically about 6' = tt/2. Consequently on the average
we have co = yco '.* With this substitution for co' in (15.29) we obtain the
radiation cross section in the laboratory:
The only change from the nonrelativistic result (15.18) is the factor y 2 in
the argument of the logarithm. Conservation of energy requires that this
* This result can be obtained from the original transformation (11.37), a>' =
yco(l — j8 cos 8), by noting that, for y ;> 1 and 6 <: 1, we have co' ~ (co/2y)(l + y 2 6 2 ).
Since the average value of y 2 6 2 in the laboratory is of the order of unity, we obtain
co' ~ co/y.
516 Classical Electrodynamics
expression be used only for frequencies such that < two < (y — \)Mc 2
^ yMc 2 . We note that quanta with laboratory energies in the range
Mc 2 < hoi < yMc 2 come from quanta with ha>' < Mc 2 in the transformed
frame K'.
The above derivation of x(°>) in the laboratory is somewhat casual in that the
dependence of transformed frequency on angle was not treated rigorously. We
should actually consider the differential cross section in the frame K':
dCl'
UyMv 2 \ "
I hot' J _
— (1 + cos 2 0')
I6rr
(15.33)
where A is the coefficient of the logarithm in (15.32). The squarebracketed
angular factor comes from the sum of the two terms in (15.9) and is normalized
to a unit integral over solid angles. When transformed according to (11.38),
(15.33) becomes in the laboratory
*>  Al, j 2 ": W ,Jl *' + £*? (15.34)
dQ. \H(o(l +y 2 d 2 )j2n (1 + y 2 2 ) 4
The angular distribution is peaked sharply in the forward direction. The angular
factor falls off as (yd)* for yd > 1. Of course, (15.34) is not valid for angles
0^1. But the order of magnitude is correct, the intensity of radiation being a
factor y~ 4 smaller at backward angles than in the forward direction, and
approaching the limiting value (at = n) :
,. d X ((o) 3 A^ UMv 2 \ , 1cr , c .
Since almost all the radiation is confined to angles 6 <; 1 , we may approximate
the solid angle element dQ, ~ 2tt0 dd = (w/y 2 ) d(y 2 6 2 ), and integrate over the
interval < y 2 2 < oo with little error. This yields the total radiation cross
section,
which differs insignificantly from the previous result (15.32).
(15.36)
15.4 Screening Effects; Relativistic Radiative Energy Loss
In the treatment of bremsstrahlung so far we have ignored the effects of
the atomic electrons. As direct contributors to the acceleration of the
incident particle they can be safely ignored, since their contribution per
atom is of the order of Z _1 times the nuclear one. But they have an indirect
effect through their screening of the nuclear charge. The potential energy
of the incident particle in the field of the atom can be approximated by the
form (13.94). This means that there will be negligible radiation emitted
for collisions at impact parameters greater than the atomic radius (13.95).
[Sect. 15.4] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 517
We can include this approximately in our previous calculations by denning
a maximum impact parameter due to screening by the atomic electrons,
b&*~a~lAJ& (15.37)
Then we must use the smaller of the two values (15.28) and (15.37) for
6max in the argument of the logarithm. The ratio is
frmax _ 192 /v\ hoi (15 38)
frmax~ZMc/yW
where m is the electronic mass and we have used the average value
oi = (x>\y. This shows that for low enough frequencies the screening value
is always smaller than (15.28). The limiting frequency co s below which we
must use 6^ ax is, in units of the spectrum end point w m ax = (y — l)Mc 2 ,
Wmax
■ ?L(»\ y lz±lY ^
192 \M/\y  1/
2Z H m /c\
192 M\vJ
Z m
192 M 7
(15.39)
where the upper (lower) line is the nonrelativistic (relativistic) limiting
form. When a> < co s the argument of the logarithm in the radiation cross
section (15.32) becomes independent of frequency:
*as~jl»2*!H (15.40)
6min Z m C
This makes the radiation cross section approach the constant value,
for co < w s . Then the energy radiated per unit frequency interval at low
frequencies is finite, rather than logarithmically divergent. This is the same
type of effect as the screening produces in making the smallangle scattering
(13.96) finite, rather than divergent as 0~ 4 for a pure Coulomb field.
Except for extremely low velocities the screening frequency co s is very
small compared to eo ma x in the nonrelativistic limit. A typical figure is
s /fi) m3X ~ 0.07 for electrons of 100Kev kinetic energy incident on a gold
target (Z = 79). For heavier nonrelativistic particles the ratio is even
smaller. This means that the spectrum shown in Fig. 15.3 is altered only
at very low frequencies for nonrelativistic bremsstrahlung.
518 Classical Electrodynamics
Fig. 15.5 Radiation cross section
in the complete screening limit.
The constant value is the semi
classical result. The curve marked
"BetheHeitler" is the quantum
mechanical Born approximation.
For extremely relativistic particles the screening can be "complete."
Complete screening occurs when co s > eomax. This occurs at energies
greater than the critical value,
_ /192M
E s=\^r 1 ) Mc ( 15  42 >
For electrons, E s ~ 42 Mev in aluminum (Z = 13) and 23 Mev in lead
(Z = 82). The corresponding values for mu mesons are 2 x 10 6 Mev and
10 6 Mev. Because of the factor Mjm, screening is important only for
electrons. When E > E s , the radiation cross section is given by the
constant value (15.41) for all frequencies. Figure 15.5 shows the radiation
cross section (15.41) in the limit of complete screening, as well as the
corresponding BetheHeitler result. Their proper quantum treatment
involves a slowly varying factor which changes from unity at co = to
0.75 at co = wmax. For cosmicray electrons and for electrons from most
highenergy electron accelerators, the bremsstrahlung is in the complete
screening limit. Thus the photon spectrum shows a typical (/zco) 1
behavior.
The radiative energy loss was considered in the nonrelativistic limit in
Section 15.2 and was found to be negligible compared to the energy loss
by collisions. For ultrarelativistic particles, especially electrons, this is no
longer true. The radiative energy loss is given approximately in the limit
y > 1 by
d£rad
dx
^i N m^\[ M "\j^\ dm (15 .43)
3 c \Mc 2 / Jo \b m J
where the argument of the logarithm depends on whether co < a> s or
ft» > co s . For negligible screening (co s < com&x) we find approximately
WW
^_^ Ar ^ L( ^_  ln(Ay)yMc2 (1544)
dx 3 nc
[Sect. 15.4] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 519
For higher energies where complete screening occurs this is modified to
dx
L6 N 5V(f!fl) 2 ln (^M)l yM ^ (15.45)
.3 He \McV \Z A m) V
showing that eventually the radiative loss is proportional to the particle's
energy.
The comparison of radiative loss to collision loss now becomes
( l\92M \
iE™>~± 111) n n \z 1A ™> (15 46)
d£coii 37r\137/M \nB a
The value of y for which this ratio is unity depends on the particle and on
Z. For electrons it is y ~ 200 for air and y ^ > 20 for lead. At higher
energies, the radiative energy loss is larger than the collision loss and for
ultrarelativistic particles is the dominant loss mechanism.
At energies where the radiative energy loss is dominant the complete
screening result (15.45) holds. Then it is useful to introduce a unit of
length X Q , called the radiation length, which is the distance a particle travels
while its energy falls to e _1 of its initial value. By conservation of energy,
we may rewrite (15.45) as
d_E__E_
dx X
with solution, E(x) = E e~ x,x ° (15.47)
where the radiation length X is
X n =
_ 3 he \Mc 2 ) n I
X192M
Z Vz m
(15.48)
For electrons, some representative values of X are 32 gm/cm 2 (270 meters)
in air at NTP, 19 gm/cm 2 (7.2 cm) in aluminum, and 4.4 gm/cm 2 (0.39 cm)
in lead.* In studying the passage of cosmicray or manmade highenergy
particles through matter, the radiation length X is a convenient unit to
employ, since not only the radiative energy loss is governed by it, but also
the production of negatonpositon pairs by the radiated photons, and so
the whole development of the electronic cascade shower.
* These numerical values differ by <~2030 per cent from those given by Rossi, p. 55,
because he uses a more accurate coefficient of 4 instead of 16/3 and Z(Z +1) instead of
Z 2 in (15.48).
520
Classical Electrodynamics
15.5 WeizsackerWilliauis Method of Virtual Quanta
The emission of bremsstrahlung and other processes involving the
electromagnetic interaction of relativistic particles can be viewed in a way
that is very helpful in providing physical insight into the processes. This
point of view is called the method of virtual quanta. It exploits the similarity
between the fields of a rapidly moving charged particle and the fields of a
pulse of radiation (see Section 11.10) and correlates the effects of the
collision of the relativistic charged particle with some system with the
corresponding effects produced by the interaction of radiation (the virtual
quanta) with the same system. The method was developed independently
by C. F. Weizsacker and E. J. Williams in 1934.
In any given collision there are an "incident particle" and a "struck
system." The perturbing fields of the incident particle are replaced by an
equivalent pulse of radiation which is analyzed into a frequency spectrum
of virtual quanta. Then the effects of the quanta (either scattering or
absorption) on the struck system are calculated. In this way the charged
particle interaction is correlated with the photon interaction. The table
lists a few typical correspondences and specifies the incident particle and
Correspondences between charged particle interactions and photon interactions
Incident Struck
Particle Process Particle System Radiative Process bmm
Bremsstrahlung in
electron (light
particle)nucleus
collision
Nucleus
Electron
(light
particle)
Scattering of virtual hjMv
photons of nuclear
Coulomb field by the
electron (light particle)
Collisional ioniza
tion of atoms (in
distant collisions)
Incident
particle
Atom
Photoejection of atomic a
electrons by virtual
quanta
Electron disintegra
tion of nuclei
Electron
Nucleus
Photodisintegration
of nuclei by virtual
quanta
Larger
of
Production of pions
in electronnuclear
collisions
Electron
Nucleus
Photoproduction of
pions by virtual
quanta interactions
with nucleus
*■ hlymv
and
R
struck system. From the table we see that the struck system is not always
the target in the laboratory. For bremsstrahlung the struck system is the
[Sect. 15.5] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 521
Pi
>■
I I
P 2
*3
Fig. 15.6 Relativistic charged particle passing the struck system S and the equivalent
pulses of radiation.
lighter of the two collision partners, since its radiation scattering power is
greater. For bremsstrahlung in electronelectron collision it is necessary
from symmetry to take the sum of two contributions where each electron
in turn is the struck system at rest initially in some reference frame.
The chief assumption in the method of virtual quanta is that the effects
of the various frequency components of equivalent radiation add inco
herently. This will be true provided the perturbing effect of the fields is
small, and is related to our assumption in Section 15.2 that the struck
particle moves only slightly during the collision.
The spectrum of equivalent radiation for an incident particle of charge
q, velocity v ~ c, passing a struck system S at impact parameter b, can be
found from the fields of Section 11.10:
yb
£ i (f) V+yW)
B 2 (t) = 0^(0
£ 3 (0 = q
yvt
(15.49)
(b 2 + y W)" J
For j8 ~ 1 the fields JE^f) and B 2 {t) are completely equivalent to a pulse of
plane polarized radiation P x incident on S in the x 3 direction, as shown in
Fig. 15.6. There is no magnetic field to accompany E 3 (t) and so form a
proper pulse of radiation P 2 incident along the x x direction, as shown.
Nevertheless, if the motion of the charged particles in S is nonrelativistic
in this coordinate frame, we can add the necessary magnetic field to create
the pulse P 2 without affecting the physical problem because the particles
in S respond only to electric forces. Even if the particles in S are influenced
by magnetic forces, the additional magnetic field implied by replacing E 3 (t)
by the radiation pulse P 2 is not important, since the pulse P 2 will be seen
to be of minor importance anyway.
522
Classical Electrodynamics
From the discussion of Section 14.5, especially equations (14.51), (14.52),
and (14.60), it is evident that the equivalent pulse P x has a frequency
spectrum (energy per unit area per unit frequency interval) I x {oj, b) given
by
Jifo b) = ± lE^atf
(15.50)
where E x (co) is the Fourier transform (14.54) of E\(i) in (15.49). Similarly
the pulse P z has the frequency spectrum,
I 2 (co, b) = ± \E 3 (oj)\*
2tt
(15.51)
The Fourier integrals have already been calculated in Chapter 13 and are
given by (13.29) and (13.30). The two frequency spectra are
Ii(o), b)
I 2 (co, b)
W
77 C \V
oM 2 K 2 (o>b
yvi \yv.
1 (2*\* ,/«*
y \yv
yv
(15.52)
We note that the intensity of the pulse P 2 involves a factor y~ 2 and so is of
little importance for ultrarelativistic particles. The shapes of these spectra
are shown qualitatively in Fig. 15.7. The behavior is easily understood if
one recalls that the fields of pulse P x are bellshaped in time with a width
At ~ b/yv. Thus the frequency spectrum will contain all frequencies up
to a maximum of order co ma x ~ 1/A*. On the other hand, the fields of
pulse P 2 are similar to one cycle of a sine wave of frequency co ~ yv/b.
1 <7 2 / \'
2 Jl
b 2
Fig. 15.7 Frequency spectra of the two equivalent pulses of radiation.
[Sect. 15.5] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 523
Consequently its spectrum will contain only a narrow range of frequencies
centered around yvjb.
In collision problems we must sum the frequency spectra (15.52) over
the various possible impact parameters. This gives the energy per unit
frequency interval present in the equivalent radiation field. As always in
such problems we must specify a minimum impact parameter 6 min . The
method of virtual quanta will be useful only if b min can be so chosen that
for impact parameters greater than Z> min the effects of the incident particle's
fields can be represented accurately by the effects of equivalent pulses of
radiation, while for small impact parameters the effects of the particle's
fields can be neglected or taken into account by other means. Setting aside
for the moment how we choose the proper value of 6 m i n , we can write
down the frequency spectrum integrated over possible impact parameters,
J(o>) = 2tt\ [I^co, b) + I 2 (a), b)]b db (15.53)
•'bmin
where we have combined the contributions of pulses P x and P 2 . This
integral has already been done in Section 13.3, equation (13.35). The
result is
7T C W
» 2
xKJx)K x {x)  fj x\K x \x) ~ K \x))
2d 1
(15.54)
where
x = ^^ (15.55)
yv
For low frequencies (co < yvjbmi^) the energy per unit frequency interval
reduces to
i^^VL^ll] (15.56,
TV C W L \ COOmin ' 2c J
whereas for high frequencies (co > yv/bmin) the spectrum falls off
exponentially as
/(co)  ~( £ ) 2 (l  —)e i2<obmiM (15.57)
Figure 15.8 shows an accurate plot of /(w) (15.54) for v ~ c, as well as the
lowfrequency approximation (15.56). We see that the energy spectrum
consists predominantly of lowfrequency quanta with a tail extending up to
frequencies of the order of lyv/bmin.
524
/(o>)
q 2 /irc
Classical Electrodynamics
— i 1 1 — i — i — tt~t
1 
0.1
Lowfrequency
approximation
j i i '■' ' '
0.5
Fig. 15.8 Frequency spectrum of virtual quanta for a relativistic particle, with the
energy per unit frequency I(a>) in units of q 2 lrrc and the frequency in units of yv/b min .
The number of virtual quanta per unit energy interval is obtained by dividing by h 2 co.
The number spectrum of virtual quanta N(hco) is obtained by using the
relation,
I(co) dco = hctiN(hco) d(H(o) (15.58)
Thus the number of virtual quanta per unit energy interval in the low
frequency limit is
„2
tt\Hc' W HcoL \ cobmin J 2c
(15.59)
The choice of minimum impact parameter & min must be considered. In
bremsstrahlung, b m \n = hjMv, where M is the mass of the lighter particle,
as discussed in Section 15.3. For collisional ionization of atoms, 6min c^ a,
the atomic radius, closer impacts being treated as collisions between the
incident particle and free electrons. In electron disintegration of nuclei or
electron production of mesons from nuclei, b m ^ = hjyMv or 6 m in = R,
the nuclear radius, whichever is larger. The values are summarized in the
table on p. 520.
[Sect. 15.6] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 525
15.6 Bremsstrahlung as the Scattering of Virtual Quanta
The emission of bremsstrahlung in a collision between an incident
relativistic particle of charge ze and mass M and an atomic nucleus of
charge Ze can be viewed as the scattering of the virtual quanta in the
nuclear Coulomb field by the incident particle in the coordinate system K',
where the incident particle is at rest. The spectrum of virtual quanta I(co')
is given by (15.54) with q = Ze. The minimum impact parameter is HjMv,
so that the frequency spectrum extends up to co' ~ yMc 2 fh.
The virtual quanta are scattered by the incident particle (the struck
system in K') according to the Thomson cross section (14.105) at low
frequencies and the KleinNishina formula (14.106) at photon energies
hco' > Mc 2 . Thus, in the frame K', for frequencies small compared to
Mc 2 /h, the radiation cross section x'(co') is given by
Z'M^f(g)V) (1560)
Since the spectrum of virtual quanta extends up to yMc 2 /h, the approxi
mation (15.56) can be used for I(co') in the region (co' < Mc 2 \h). Thus the
radiation cross section becomes
Ky ' 3 c \Mc 2 ' L \ hco' J
(15.61)
where extreme relativistic motion (v ^ c) has been assumed.
This is essentially the same cross section as (15.29), and can be trans
formed to the laboratory in the same way as was done in Section 15.3.
Equations (15.60) and (15.61), involving the Thomson cross section, are
valid only for quanta in K' with frequencies co' < Mc 2 jh. For frequencies
co' > Mc 2 \h, we must replace the constant Thomson cross section (14.105)
with the quantummechanical KleinNishina formula (14.106), which falls
off rapidly with increasing frequency. This shows that in K' the bremsstrah
lung quanta are confined to a frequency range < co' < Mc 2 /h, even
though the spectrum of virtual quanta in the nuclear Coulomb field
extends to much higher frequencies. The restricted spectrum in K' is
required physically by conservation of energy, since in the laboratory
system where co = yco' the frequency spectrum is limited to < co <
(yMc 2 /h). A detailed treatment using the angular distribution of scattering
from the KleinNishina formula yields a bremsstrahlung cross section in
complete agreement with the BetheHeitler formulas (Weizsacker, 1934).
The effects of screening on the bremsstrahlung spectrum can be dis
cussed in terms of the Weizsacker Williams method. For a screened
526 Classical Electrodynamics
Coulomb potential the spectrum of virtual quanta is modified from (15.56).
The argument of the logarithm is changed to a constant, as was discussed
in Section 15.4.
Further applications of the method of virtual quanta to such problems
as collisional ionization of atoms and electron disintegration of nuclei are
deferred to the problems at the end of the chapter.
15.7 Radiation Emitted during Beta Decay
In the process of beta decay an unstable nucleus with atomic number Z
transforms spontaneously into another nucleus of atomic number (Z ± 1)
while emitting an electron (=fe) and a neutrino. The process is written
symbolically as
Z>{Z± l) + e* +v (15.62)
The energy released in the decay is shared almost entirely by the electron
and the neutrino, with the recoiling nucleus getting a completely negligible
share because of its very large mass. Even without knowledge of why or
how beta decay takes place, we can anticipate that the sudden creation of a
rapidly moving charged particle will be accompanied by the emission of
radiation. As mentioned in the introduction, either we can think of the
electron initially at rest and being accelerated violently during a short time
interval to its final velocity, or we can imagine that its charge is suddenly
turned on in the same short time interval. The heavy nucleus receives a
negligible acceleration and so does not contribute to the radiation.
For the purposes of calculation we can assume that at t = an electron
is created at the origin with a constant velocity v = c$. Then, according
to (14.67), the intensity distribution in frequency and angle of the radiation
emitted is
dI(co) e 2 co 2 f°° , ^ i0 >(t±™\ , 2
v  In x (n x P)e*T « ) dt
Jo
dO.
\tt*C
(15.63)
Since (5 is constant, the position of the electron is r(t) = c$t. Then the
intensity distribution is
^ = £^> sin 2 r
dQ. 477V Jo
yico(l — p cos 6)t
dt
(15.64)
where 6 is measured from the direction of motion of the emerging electron.
Thus the angular distribution is
dljoi) = _£!_ 02 sin 2
dQ 4tt 2 c P (1  cos Of U >
[Sect. 15.7] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 527
v(t)
Fig. 15.9
while the total intensity per unit frequency interval is
/(co) = 
7TC
A In
18/ J
(15.66)
For 8 < 1, (15.66) reduces to /(co) ^± 2e 2 8 2 /37rc, showing that for low
energy beta particles the radiated intensity is negligible.
The intensity distribution (15.66) is a typical bremsstrahlung spectrum
with number of photons per unit energy range given by
N(ha))
= _e 2 _/j_
irhc \haiJ LB
±ln
1 +
1 
2
(15.67)
It sometimes bears the name "inner bremsstrahlung" to distinguish it from
bremsstrahlung emitted by the same beta particle in passing through
matter. It appears that the spectrum extends to infinity, thereby violating
conservation of energy. We can obtain qualitative agreement with con
servation of energy by appealing to the uncertainty principle. Figure 15.9
shows a qualitative sketch of the electron velocity as a function of time.
Our calculation is based on a step function with the acceleration time t
vanishingly small. From the uncertainty principle, however, we know that
for a given uncertainty in energy AE the uncertainty in time At cannot be
smaller than At ~ fi/AE. In the act of creation of the beta particle,
AE = E = ymc 2 , so that the acceleration time t must be of the order of
t ^ fr/E. When this is transformed into frequency, the wellknown
arguments show that the frequency spectrum will not extend appreciably
beyond eo max ~ E\h, thereby satisfying the conservation of energy require
ment at least qualitatively.
The total energy radiated is approximately
£rad
Jo
I{pS) da> ^
e
irhc
.8 \\BJ J
(15.68)
528 Classical Electrodynamics
For very fast beta particles, the ratio of energy going into radiation to the
particle energy is
Erad ^ 2 e
2
\mcv
(15.69)
This shows that the radiated energy is a very small fraction of the total
energy released in beta decay, even for the most energetic beta processes
Cdnax ^ 30mc 2 ). Nevertheless, the radiation can be observed, and pro
vides useful information for nuclear physicists.
In the actual beta process the energy release is shared by the electron
and the neutrino so that the electron has a whole spectrum of energies up
to some maximum. Then the radiation spectrum (15.66) must be averaged
over the energy distribution of the beta particles. Furthermore, a quantum
mechanical treatment leads to modifications near the upper end of the
photon spectrum. These are important details for quantitative comparison
with experiment. But the origins of the radiation and its semiquantitative
description are given adequately by our classical calculation.
15.8 Radiation Emitted in OrbitalElectron Capture— Disappearance of
Charge and Magnetic Moment
In beta emission the sudden creation of a fast electron gives rise to
radiation. In orbitalelectron capture the sudden disappearance of an
electron does likewise. Orbitalelectron capture is the process whereby an
orbital electron around an unstable nucleus of atomic number Z is
captured by the nucleus, transforming it into another nucleus with atomic
number (Z — 1), with the simultaneous emission of a neutrino which
carries off the excess energy. The process can be written symbolically as
Z + e*(Zl) + v (15.70)
Since a virtually undetectable neutrino carries away the decay energy if
there is no radiation, the spectrum of photons accompanying orbital
electron capture is of great importance in yielding information about the
energy release.
As a simplified model we consider an electron moving in a circular
atomic orbit of radius a with a constant angular velocity co . The orbit lies
in the xy plane, as shown in Fig. 15.10, with the nucleus at the center. The
observation direction n is defined by the polar angle and lies in the xz
plane. The velocity of the electron is
\(t) = — €l co a sin (a> t + a) + € 2 co a cos (co t + a) (15.71)
where a is an arbitrary phase angle. If the electron vanishes at / = 0, the
[Sect. 15.8] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 529
Fig. 15.10
frequency spectrum of emitted radiation (14.67) is approximately
dljco)
dQ.
e 2 co 2
4tt 2 c 3
I ° n x [n x v(0y°* dt
J — CO
(15.72)
where we have assumed that jcoajc) < 1 (dipole approximation) and put
the retardation factor equal to unity. The integral in (15.72) can be
written /.„
dt = — o> a(€ 1 / 1 + € M cos 0/ 2 ) (15.73)
J — oo
where ,.„
I x = cos (ay + 0L)e i<ot dt
J — 00
f
J —co
sin jco t + a)e i0,t df
(15.74)
and € ± , € are unit polarization vectors perpendicular and parallel to the
plane containing n and the z axis. The integrals are elementary and lead
to an intensity distribution,
dI(co) e 2 co 2
2^2
a>n a
dQ. 4tt 2 c 3 (a) 2  co 2 f
X [(o> 2 cos 2 a + w 2 sin 2 a) + cos 2 d(co 2 sin 2 a + co 2 cos 2 a)] (15.75)
Since the electron can be captured from any position around the orbit, we
average over the phase angle a. Then the intensity distribution is
dljco)
dQ
The total energy radiated per unit frequency interval is
i) = _il M* coW + O K1 + cos 2 d) (15 76)
! 4tt 2 c A c J (co 2 — oo 2 f
I(fi>)
— 2 e 2 /co aY
3rr c \ ci
' oo\(x> 2 + co 2 )"
. (a, 2  co 2 ) 2 J
(15.77)
530
Classical Electrodynamics
Nftco)
Fig. 15.11 Spectrum of photons emitted in orbital electron capture because of dis
appearance of the charge of the electron.
while the number of photons per unit energy interval is
O} 2 ((O 2 + co 2 )
*>£©(?)'
(co 2  co 2 )
2\2
1
hco
— (15.78)
For co > co the squarebracketed quantity approaches unity. Then
the spectrum is a typical bremsstrahlung spectrum. But for co ~ co the
intensity is very large (infinite in our approximation). The behavior of the
photon spectrum is shown in Fig. 15.11. The singularity at co = co may
seem alarming, but it is really quite natural and expected. If the electron
were to keep orbiting forever, the radiation spectrum would be a sharp line
at co = co . The sudden termination of the periodic motion produces a
broadening of the spectrum in the neighborhood of the characteristic
frequency.
Quantum mechanically the radiation arises when an / = 1 electron
(mainly from the 2p orbit) makes a virtual radiative transition to an / =
state, from which it can be absorbed by the nucleus. Thus the frequency
co must be identified with the frequency of the characteristic 2p > \s
Xray, hco ~ (3Z 2 e 2 /$a ). Similarly the orbit radius is actually a transitional
dipole moment. With the estimate a ~ a /Z, where a is the Bohr radius,
the photon spectrum (15.78) is
Mto^Lz^YLrNV + «*">'
2>2tt \hc/ Hco
L (co 2  co 2 f
(15.79)
[Sect. 15.8] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 531
The essential characteristics of this spectrum are its strong peaking at the
Xray energy and its dependence on atomic number as Z 2 .
So far we have considered the radiation which accompanies the disap
pearance of the charge of an orbital electron in the electroncapture
process. An electron possesses a magnetic moment as well as a charge.
The disappearance of the magnetic moment also gives rise to radiation,
but with a spectrum of quite different character. The intensity distribution
in angle and frequency for a point magnetic moment in motion is given by
(14.74). The electronic magnetic moment can be treated as a constant
vector in space until its disappearance at t = 0. Then, in the dipole
approximation, the appropriate intensity distribution is
dIM = J^L I f ° n x ^ dt 2 (15.80)
dQ 47rVU°o
This gives
^) = ^L„ 2 sin 2 (15.81)
dQ 47T 2 c sr
where is the angle between jjl and the observation direction n.
In a semiclassical sense the electronic magnetic moment can be thought
of as having a magnitude fx = V3(ehl2mc), but being observed only
through its projection p z = ±{eh\2mc) on an arbitrary axis. The moment
can be thought of as precessing around the axis at an angle a = tan 1 V 2,
so that on the average only the component of the moment along the axis
survives. It is easy to show that on averaging over this precession sin 2
in (15.81) becomes equal to its average value of f , independent of obser
vation direction. Thus the angular and frequency spectrum becomes
dlioy) = _^_/M 2 (1582)
dQ. 877 2 cW 2 /
The total energy radiated per unit frequency interval is
/(„) = f{^X (15.83)
2nc \mc 2 '/
while the corresponding number of photons per unit energy interval is
iV(M = r4^r 2 (1584)
IttHc (mc 2 f
These spectra are very different in their frequency dependence from a
bremsstrahlung spectrum. They increase with increasing frequency,
apparently without limit. Of course, we have been forewarned that our
532
Classical Electrodynamics
classical results are valid only in the lowfrequency limit. We can imagine
that some sort of uncertaintyprinciple argument such as was used in
Section 15.7 for radiative beta decay holds here and that conservation of
energy, at least, is guaranteed. Actually, modifications arise because a
neutrino is always emitted in the electroncapture process. The probability
of emission of the neutrino can be shown to depend on the square of its
energy E v . When no photon is emitted, the neutrino has the full decay
energy E v = E . But when a photon of energy hco accompanies it, the
neutrino's energy is reduced to EJ = E — hco. Then the probability of
neutrino emission is reduced by a factor,
eA 2 = L _ HcoY
Ej \ EJ
(15.85)
This means that our classical spectra (15.83) and (15.84) must be corrected
by multiplication with (15.85) to take into account the kinematics of the
neutrino emission. The modified classical photon spectrum is
N(hco) =
ha>
2\2
Inhc (mc )
. hco
(15.86)
This is essentially the correct quantummechanical result. A comparison
of the corrected distribution (15.86) and the classical one (15.84) is shown
in Fig. 15.12. Evidently the neutrinoemission probability is crucial in
obtaining the proper behavior of the photon energy spectrum. For the
N(hu)
Fig. 15.12 Spectrum of photons emitted in orbital electron capture because of dis
appearance of the magnetic moment of the electron.
[Sect. 15.8] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 533
N(fio>)
hoiQ
Fig. 15.13 Typical photon spectrum for radiative orbital electron capture with energy
release E , showing the contributions from the disappearance of the electronic charge
and magnetic moment.
customary bremsstrahlung spectra such correction factors are less im
portant because the bulk of the radiation is emitted in photons with
energies much smaller than the maximum allowable value.
The total radiation emitted in orbital electron capture is the sum of the
contributions from the disappearance of the electric charge and of the
magnetic moment. From the different behaviors of (15.79) and (15.86) we
see that the upper end of the spectrum will be dominated by the magnetic
moment contribution, unless the energy release is very small, whereas the
lower end of the spectrum will be dominated by the electriccharge term,
especially for high Z. Figure 15.13 shows a typical combined spectrum
for Z ~ 2030. Observations on a number of nuclei confirm the general
features of the spectra and allow determination of the energy release
E n .
REFERENCES AND SUGGESTED READING
Classical bremsstrahlung is discussed briefly by
Landau and Lifshitz, Classical Theory of Fields, Section 9.4,
Panofsky and Phillips, Section 19.6.
A semiclassical discussion, analogous to ours, but much briefer, appears in
Rossi, Section 2.12.
534 Classical Electrodynamics
Bremsstrahlung can be described accurately only by a proper quantummechanical
treatment. The standard reference is
Heitler.
The method of virtual quanta (Weizsacker Williams method) has only one proper
reference, the classic article by
Williams.
Short discussions appear in
Heitler, Appendix 6,
Panofsky and Phillips, Section 18.5.
Among the quantummechanical treatments of radiative beta processes, having com
parisons with experiment in some cases, are those by
C. S. W. Chang and D. L. Falkoff, Phys. Rev., 76, 365 (1949),
P. C. Martin and R. J. Glauber, Phys. Rev., 109, 1307 (1958),
Siegbahn, Chapter XX (III) by C. S. Wu.
PROBLEMS
15.1 A nonrelativistic particle of charge e and mass m collides with a fixed,
smooth, hard sphere of radius R. Assuming that the collision is elastic,
show that in the dipole approximation (neglecting retardation effects) the
classical differential cross section for the emission of photons per unit
solid angle per unit energy interval is
d 2 o R* e 2/ y \2 j
 diU:(2+3sin«e)
dQ.d(ft<jL>) I2n hc\cj fra>
where B is measured relative to the incident direction. Sketch the angular
distribution. Integrate over angles to get the total bremsstrahlung cross
section. Qualitatively what factor (or factors) govern the upper limit to
the frequency spectrum ?
15.2 Two particles with charges q 1 and q 2 and masses m x and m 2 collide under the
action of electromagnetic (and perhaps other) forces. Consider the angular
and frequency distributions of the radiation emitted in the collision.
(a) Show that for nonrelativistic motion the energy radiated per unit
solid angle per unit frequency interval in the center of mass coordinate
system is given by
dI(io, fi) n*
dn 4t7 2 c 3
J \rrti m 2 J
where x = (x 2 — x 2 ) is the relative coordinate, n is a unit vector in the
direction of observation, and fi = m x m 2 \(m x + m 2 ) is the reduced mass.
(b) By expanding the retardation factors show that, if the two particles
have the same charge to mass ratio (e.g., a deuteron and an alpha particle),
[Probs. 15] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes
the leading (dipole) term vanishes and the nextorder term gives
dI{co, Q) <*{f£ + qg£t I Li„ Kn . x)( x x n) dt 2
535
dCl
4^
(c) Relate result (6) to the multipole expansion of Sections 9.19.3.
15.3 Two identical point particles of charge q and mass m interact by means of
a shortrange repulsive interaction which is equivalent to a hard sphere of
radius R in their relative separation. Neglecting the electromagnetic inter
action between the two particles, determine the radiation cross section in the
center of mass system for a collision between these identical particles to the
lowest nonvanishing approximation. Show that the differential cross section
for emission of photons per unit solid angle per unit energy interval is
d 2 a
d(haj) dQ.
= §M^ hoi{l
+ iWcos 0)   2 V 4 (cos 0)]
where is measured relative to the incident direction. Compare this result
with that of Problem 15.1 as to frequency dependence, relative magnitude,
etc.
15.4 A particle of charge ze, mass m, and nonrelativistic velocity v is deflected in
a screened Coulomb field, V(r) = Zze 2 e~ ar lr, and consequently emits
radiation. Discuss the radiation with the approximation that the particle
moves in a straightline trajectory past the force center.
(a) Show that, if the impact parameter is b, the energy radiated per unit
frequency interval is
I{o>, b) = 1 — s 1 1 — 1 a^VO
3tt c \mc i ] \v/
for to < v/b, and negligible for co > vjb.
(b) Show that the radiation cross section is
, s 16 ZV/zVV/c
x(co) ^T—\7n7>)\v
2r
■(k \x)KJ
\x) +
IK^Kjix)
Jx 1
where X x = a&min, X 2 = 0C& m ax.
(c) With bmm = fi/mv, 6 max = i?/o>, and or 1 = l.4a Z~^, determine the
radiation cross section in the two limits, x 2 <^l an(i *2 ^ * • Compare your
results with the "screening" and "no screening" limits of the text.
15.5 A particle of charge ze, mass m, and velocity v is deflected in a hyperbolic
path by a fixed repulsive Coulomb potential, V(r) = Zze 2 jr. In the non
relativistic dipole approximation (but with no further approximations),
(a) show that the energy radiated per unit frequency interval by the
particle when initially incident at impact parameter b is :
8 (zeaco) 2 , ,
' o) (h^Q_
3tt c 3
(b) show that the radiation cross section is
16 {zeavf
+
 1
K.
i(o/ca
Z()= T ^
g— (JTOJ/CO
CO / 0)\ , I U>\
a>o Wo/ L \ w o/
536 Classical Electrodynamics
(c) Prove that the radiation cross section reduces to that obtained in the
text for classical bremsstrahlung for co < a> . What is the limiting form
fora>>co ? b
(d) What modifications occur for an attractive Coulomb interaction?
The hyperbolic path may be described by
x = a( e + cosh £), y = b sinh £, to t = (f + e sinh f)
where a = Zze 2 /mv 2 , e = Vl + (b/a) 2 , co = vja.
15.6 Using the method of virtual quanta, discuss the relationship between the
cross section for photodisintegration of a nucleus and electrondisintegration
of a nucleus.
(a) Show that, for electrons of energy E = ymc 2 > mc 2 , the electron
disintegration cross section is approximately:
where hw> T is the threshold energy for the process.
(b) Assuming that ff photo (eo) has the resonance shape:
A e 2 r
"photoC") ^ Yn Mc (coco^+cr^
where the width T is small compared to (co  co T ), sketch the behavior of
o e i(E) as a function of E and show that for E > fico ,
a el (E) ~  () — 1 In / ^ 2 \
v \HcJ Mc co \mc 2 ho) J
(c) Discuss the experimental comparison between activities produced by
bremsstrahlung spectra and monoenergetic electrons as presented by
Brown and Wilson, Phys. Rev., 93, 443 (1954), and show that the quantity
defined as F e x P (Z, E) has the approximate value 8tt/3 at high energies if the
Weizsacker Williams spectrum is used to describe both processes and the
photodisintegration cross section has a resonance shape.
15.7 A fast particle of charge ze, mass M, velocity v, collides with a hydrogenlike
atom with one electron of charge — e, mass m, bound to a nuclear center of
charge Ze. The collisions can be divided into two kinds: close collisions
where the particle passes through the atom (b < d), and distant collisions
where the particle passes by outside the atom (b > d). The atomic "radius"
d can be taken as a /Z. For the close collisions the interaction of the
incident particle and the electron can be treated as a twobody collision and
the energy transfer calculated from the Rutherford cross section. For the
distant collisions the excitation and ionization of the atom can be considered
the result of the photoelectric effect by the virtual quanta of the incident
particle's fields.
For simplicity assume that for photon energies Q greater than the
ionization potential / the photoelectric cross section is
[Probs. 15] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 537
(This obeys the empirical Z 4 A 3 law for Xray absorption and has a coefficient
adjusted to satisfy the dipole sum rule, J o v (Q) dQ = l^e^/mc.)
(a) Calculate the cross sections for energy transfer Q for close and
distant collisions (write them as functions of Q\l as far as possible and in
units of iTTZ^Imv 2 ^). Plot the two distributions for Qjl > 1 for non
relativistic motion of the incident particle and \mv 2 = 10 3 /.
(b) Show that the number of distant collisions is much larger than the
number of close collisions, but that the energy transfer per collision is
much smaller. Show that the energy loss is divided approximately equally
between the two kinds of collisions, and verify that your total energy loss
is in essential agreement with Bethe's result (13.44).
15.8 In the decay of a pi meson at rest a mu meson and a neutrino are created.
The total kinetic energy available is (m„ — m^c 2 = 34 Mev. The mu
meson has a kinetic energy of 4.1 Mev. Determine the number of quanta^
emitted per unit energy interval because of the sudden creation of the
moving mu meson. Assuming that the photons are emitted perpendicular
to the direction of motion of the mu meson (actually it is a sin 2 distribu
tion), show that the maximum photon energy is 17 Mev. Find how many
quanta are emitted with energies greater than onetenth of the maximum,
and compare your result with the observed ratio of radiative pimu decays.
[W. F. Fry, Phys. Rev., 86, 418 (1952); H. Primakoff, Phys. Rev., 84, 1255
(1951).]
15.9 In internal conversion, the nucleus makes a transition from one state to
another and an orbital electron is ejected. The electron has a kinetic energy
equal to the transition energy minus its binding energy. For a conversion
line of 1 Mev determine the number of quanta emitted per unit energy
because of the sudden ejection of the electron. What fraction of the electrons
will have energies less than 99 per cent of the total energy? Will this
lowenergy tail on the conversion line be experimentally observable?
16
Multipole Fields
In Chapters 3 and 4 on electrostatics the spherical harmonic
expansion of the scalar potential was used extensively for problems
possessing some symmetry property with respect to an origin of coordinates.
Not only was it useful in handling boundaryvalue problems in spherical
coordinates, but with a source present it provided a systematic way of
expanding the potential in terms of multipole moments of the charge
density. For timevarying electromagnetic fields the scalar spherical
harmonic expansion can be generalized to an expansion in vector spherical
waves. These vector spherical waves are convenient for electromagnetic
boundaryvalue problems possessing spherical symmetry properties and for
the discussion of multipole radiation from a localized source distribution.
In Chapter 9 we have already considered the simplest radiating multipole
systems. In the present chapter we present a systematic development.
16.1 Basic Spherical Wave Solutions of the Scalar Wave Equation
As a prelude to the vector spherical wave problem, we consider the
scalar wave equation. A scalar field ^(x, t) satisfying the sourcefree
wave equation, 2
V V i = (16.1,
can be Fourieranalyzed in time as
/*oo
ip(x, = y(x, (o)e i(at da) (16.2)
J — oo
with each Fourier component satisfying the Helmholtz wave equation,
(V 2 + k 2 )y)(x, co) = (16.3)
538
[Sect. 16.1]
Multipole Fields
539
with k 2 = co 2 jc 2 . For problems possessing symmetry properties about
some origin it is convenient to have fundamental solutions appropriate to
spherical coordinates. The representation of the Laplacian operator in
spherical coordinates is given in equation (3.1). The separation of the
angular and radial variables follows the wellknown expansion,
<Kx, co) = 2 Mr)Y lm (0, <f>)
(16.4)
where the spherical harmonics Y lm are defined by (3.53). The radial
functions / f (r) satisfy the radial equation,
• d 2 2 d 2 /(/ + 1) '
Jr 2 rdr r 2 .
Mr) =
With the substitution,
fi(r) = rz u t (r)
equation (16.5) is transformed into
d 2 Id ; , 2 (/ + ) 2 '
Ur 2 rdr r 2 .
»i(r) =
(16.5)
(16.6)
(16.7)
This equation is just Bessel's equation (3.75) with v = I + \. Thus the
solutions for fir) are
(16.8)
fir) ~ £■ J t+ i4(kr), % N l+l4 (kr)
It is customary to define spherical Bessel and Hankel functions, denoted
by ;*(*)> «i(«), ti?' 2) {x), as follows:
nix) = \^A N l+ i A (x)
h? 2 \x) = (£) [_J l+ y 2 (x) ± iN l + H (x)]
(16.9)
For real x, h\ 2) (x) is the complex conjugate of h^ix). From the series
expansions (3.82) and (3.83) one can show that
\a; dxl \ x I
n x {x)= (*)M — I )
\x dxl \ x !
(16.10)
540 Classical Electrodynamics
For the first few values of / the explicit forms are :
JoO)
sin a;
n (x) =
cos a;
IX
i ( T \  !iBi? cosx „ r„\ cos a; sin x
h?\x) =  e (i + ')
x \ xl
j&) = (\ ~ ) sin x _ Icosf ^ {x) = _ /3 l\ cos x _ sin*
^ xl X 2 \x 3 Xl x 2
a; \ x x £ I
(16.11)
From the asymptotic forms (3.89)(3.91) it is evident that the small
argument limits are
x </
Ji( x )
(21 + 1)! !
"*(*) > 
(21  1)! !
(16.12)
where (2/ + 1)! ! = (21 + 1)(2/  1)(2/  3) • • • (5) • (3) • (1). Similarly
the large argument limits are
x >/
Ji(x)> sin [x— I
x \ 2/
n t (x) + cos he
x \ 2/
(16.13)
The spherical Bessel functions satisfy the recursion formulas
2/ + 1
— — z t (x) = z^x) + z l+1 (x)
1
Zl ' (x) = JTTi V z i^  V + i>i+i(*>D
(16.14)
[Sect. 16.1] Multipole Fields 541
where z t (x) is any one of the functions j\(x), «,(*), htpix), h\ 2) (x). The
Wronskians of the various pairs are
W(h n,) =  W(j t1 h™) =  W(n lt h™) = \ (16.15)
The general solution of (16.3) in spherical coordinates can be written
Vtt = 2 lAWXkr) + A^hf\kr)\Y lm {d, <f>) (16.16)
where the coefficients A\H and 4}^ will be determined by the boundary
conditions.
For reference purposes we present the spherical wave expansion for the
outgoing wave Green's function G(x, x'), which is appropriate to the
equation,
(V 2 + k 2 )G(x, x') = <3(x  x') (16.17)
in the infinite domain. The closed form for this Green's function, as was
shown in Chapter 9, is
ik\x*\
G(x,x') = ^ (16.18)
47t\x — x I
The spherical wave expansion for G(x, x') can be obtained in exactly the
same way as was done in Sections 3.8 and 3.10 for Poisson's equation
[see especially equation (3.117) and below, and (3.138) and below]. An
expansion of the form,
G(x, x') = 2 gfr, r')Y* m (0', <f>')Y lm (d, <f>) (16.19)
l,m
substituted into (16.17) leads to an equation for g t (r, r'):
il + I ±L + k 2 _ KijtAi
dr 2 r dr r 2
U = ~  2 Kr ~ r') (16.20)
r 2
The solution which satisfies the boundary conditions of finiteness at the
origin and outgoing waves at infinity is
gl (r, r') = AUk r< )h^(kr>) (16.21)
The correct discontinuity in slope is assured if A = ik. Thus the expansion
of the Green's function is
i]c\x—x'\ °°. I
1 = ik y j l (/cr<)^ 1) (/cr>) > Y*Jd', <f>')Y lm (6, <f>) (16.22)
4tt x — x r1 t 1 ,
542
Classical Electrodynamics
Our emphasis so far has been on the radial functions appropriate to
the scalar wave equation. We now reexamine the angular functions in
order to introduce some concepts of use in considering the vector wave
equation. The basic angular functions are the spherical harmonics
Yim(Q> 4) (353), which are solutions of the equation,
1
1
.sin dd \ 30/ sin 2 d<f> 2
Y lm =l{l + \)Y lm (16.23)
As is well known in quantum mechanics, this equation can be written in
the form :
L 2 Y lm = l{l+\)Y ln (16.24)
The differential operator L 2 = L 2 + L v 2 + L 2 , where
L =  (r x V)
(16.25)
is the orbital angularmomentum operator of wave mechanics.
The components of L can be written conveniently in the combinations,
L + =L X + iL y = <?*(— + i cot —
+ x \do dcf>
L_ = L x  iL v = e'+l — + / cot —
\ dd dcf>
r • d
defy
(16.26)
We note that L operates only on angular variables and is independent of r.
From definition (16.25) it is evident that
rL =
(16.27)
holds as an operator equation. From the explicit forms (16.26) it is easy
to verify that L 2 is equal to the operator on the left side of (16.23).
From the explicit forms (16.26) and recursion relations for Y lm the
following useful relations can be established:
L + Y lm = V(l  m)(l + m + \)Y h
L_Y lm = V(l + m)(lm + l)Y l:
w— 1
L K Y, m = mYr
(16.28)
[Sect. 16.2] Multipole Fields 543
Finally we note the following operator equations concerning the com
mutation properties of L, L 2 , and V 2 :
L 2 L = LL 2 1
LxL = /L ► (16.29)
L,.V 2 = V 2 L,
where
(16.30)
V 2 = —Ar)
rdr 2
16.2 Multipole Expansion of the Electromagnetic Fields
In a sourcefree region Maxwell's equations are
c dt c ot
(16.31)
V • E = 0, V • B =
With the assumption of a time dependence, e~ mt , these equations become
VxE = iB, V x B =  ikE
V • E = 0, V.B =
(16.32)
If E is eliminated between the two curl equations, we obtain the following
equations,
(V 2 + fc 2 )B = 0, V • B = "
and the defining relation,
E = V xB
k
Alternatively B can be eliminated to yield
(V 2 + fc 2 )E = 0, V • E =
(16.33)
plus
B = — V xE
(16.34)
Either (16.33) or (16.34) is a set of three equations which is equivalent to
Maxwell's equations (16.32).
We now wish to determine multipole solutions for E and B. From
(16.33) it is evident that each rectangular component of B satisfies the
Helmholtz wave equation (16.3). Hence each component of B can be
544 Classical Electrodynamics
represented by the general solution (16.16). These can be combined to
yield the vectorial result :
B = 1 [AgftPtyr) + A£h«>(fcr)]y IfII (0, <j>) (16.35)
l ,m
where A lm are arbitrary constant vectors.
The coefficients A lm in (16.35) are not completely arbitrary. The
divergence condition V • B = must be satisfied. Since the radial
functions are linearly independent, the condition V • B = must hold for
the two sets of terms in (16.35) separately. Thus we require the coefficients
A lm to be so chosen that
V • ^ hflcrfr^JO, <f>) = (16.36)
l,m
The gradient operator can be written in the form :
r dr
V =  r xL (16.37)
where L is the operator (16.25). When this is applied in (16.36), we obtain
the requirement,
r • 2 [f: 2 Alm Ylm  7 1 L x 2 AimYlm ] = ° (1638)
I m m
From recursion formulas (16. 14) it is evident that in general the coefficients
A lm for a given / will be coupled with those for /' = / ± 1. This will
happen unless the (2/ + 1) vector coefficients for each / value are such that
rlA ( J Im = (16.39)
TO
For this special circumstance, the second term in (16.38) shows that the
final condition on the coefficients is
r(L xZA lm Y l J = (16.40)
TO
The assumption (16.39) that the field is transverse to the radius vector,
together with (16.40), is sufficient to determine a unique set of vector
angular functions of order /, one for each m value. These can be found in
a straightforward manner from (16.39) and (16.40), and the properties of
the Y lm 's. But it is expedient, and not too damaging at this point, to
observe that the appropriate angular solution is
2 A lm' Y lm' = 2 ^ ln LY lm (16.41)
to' m
From (16.27) it is clear that the transversality condition (16.39) is satisfied.
Similarly, from the second commutation relation in (16.29) and (16.27),
[Sect. 16.2] Multipole Fields 545
the final condition (16.40) is obeyed. That the functions [/i(r)IT im ]
satisfy the wave equation (16.3) follows from the last commutation
relation in (16.29).
By assumption (16.39) we have found a special set of electromagnetic
multipole fields,
B Im =/,(fcr)Ly Im (M))
Ej m — T v X B lm
k
(16.42)
Kim = ~ V X °lm J
where
f l{ kr) = APh?\kr) + AfWfXkr) (16.43)
Any linear combination of these fields, summed over / and m, satisfies the
set of equations (16.33). They have the characteristic that the magnetic
induction is perpendicular to the radius vector (r • B lm = 0). They
therefore do not represent a general solution to equations (16.33). They
are, in fact, the spherical equivalent of the transverse magnetic (TM), or
electric, cylindrical fields of Chapter 8.
If we had started with the set of equations (16.34) instead of (16.33), we
would have obtained an alternative set of multipole fields in which E is
transverse to the radius vector:
K lm =Mkr)LY lm (0,cf>))
»« T, ( 1644 )
B Jro = — V x E fTO J
These are the spherical wave analogs of the transverse electric (TE), or
magnetic, cylindrical fields of Chapter 8.
Just as for the cylindrical waveguide case, the two sets of multipole
fields (16.42) and (16.44) can be shown to form a complete set of vector
solutions to Maxwell's equations. The terminology electric and magnetic
multipole fields will be used, rather than TM and TE, since the sources of
each type of field will be seen to be the electriccharge density and the
magneticmoment density, respectively. Since the vector spherical
harmonic, L Y lm , plays an important role, it is convenient to introduce the
normalized form,*
x lm (d, <f>) = ,i — Lr Jm (0, <f>) (16.45)
V/(/ + i)
with the orthogonality property,
X? m ..X lm <*Q = Mmm (1646)
/■
* Xi m is defined to be identically zero for / = 0. Spherically symmetric solutions to
the sourcefree Maxwell's equations exist only in the static limit k * 0.
546 Classical Electrodynamics
By combining the two types of fields we can write the general solution to
Maxwell's equations (16.32):
B = ^ [a E (l, m)fflcr)X lm  l  a M (l, m)V x g,(fcr)X,.
l,m
E = ^ Mr a E (l, m)V x /j(/cr)X ?m + a M (l, m^fciOX,
(16.47)
where the coefficients a E {l, m) and a M (I, m) specify the amounts of electric
(/, m) multipole and magnetic (/, m) multipole fields. The radial functions
fi(kr) andg;(&r) are of form (16.43). The coefficients a E (l, m) and a M (l, m),
as well as the relative proportions in (16.43), will be determined by the
sources and boundary conditions.
16.3 Properties of Multipole Fields; Energy and Angular Momentum
of Multipole Radiation
Before considering the connection between the general solution (16.47)
and a localized source distribution, we examine the properties of the
individual multipole fields (16.42) and (16.44). In the near zone (kr < 1)
the radial function f t (kr) is proportional to n^ given by (16.12), unless its
coefficient vanishes identically. Excluding this possibility, the limiting
behavior of the magnetic induction for an electric (/, m) multipole is
B lw yL^ (16.48)
where the proportionality coefficient is chosen for later convenience. To
find the electric field we must take the curl of the righthand side. A
useful operator identity is
7 2 «/. . ^
iV x L = rV 2  VI 1 + r — I (16.49)
The electric field (16.42) is
E,.yVxL(&) (16.50)
Since (Y lm lr l+1 ) is a solution of Laplace's equation, the first term in
(16.49) vanishes. The second term merely gives a factor /. Consequently
the electric field at close distances for an electric (/, m) multipole is
E lm *V ^ (16.51)
[Sect. 16.3] Multipole Fields 547
This is exactly the electrostatic multipole field of Section 4.1. We note
that the magnetic induction B lm is smaller in magnitude than E lm by a
factor kr. Hence, in the near zone, the magnetic induction of an electric
multipole is always much smaller than the electric field. For the magnetic
multipole fields (16.44) evidently the roles of E and B are interchanged
according to the transformation,
E S ^B M , B B ^E M (16.52)
In the far or radiation zone {kr > 1) the multipole fields depend on the
boundary conditions imposed. For definiteness we consider the example
of outgoing waves, appropriate to radiation by a localized source. Then
the radial function f t (kr) is proportional to the spherical Hankel function
hl x) {kr). From the asymptotic form (16.13) we see that in the radiation
zone the magnetic induction for an electric (/, m) multipole goes as
B^>(*y +1 f r L7 lrn (16.53)
kr
Then the electric field can be written
(0*
E7™ =
ikr\ p ikr
V\ e —J xLY lm + ^V xLY t1
(16.54)
Since we have already used the asymptotic form of the spherical Hankel
function, we are not justified in keeping higher powers in (1/r) than the
first. With this restriction and use of the identity (16.49) we find
E Jm = (0 l+1 f" n x LY JW  i (rV 2  V)y If
kr L k
(16.55)
where n = (r/r) is a unit vector in the radial direction. The second term is
evidently \\kr times some dimensionless function of angles and can